Bayesian GLM Part6

Author

Murray Logan

Published

07/07/2025

1 Preparations

Load the necessary libraries

library(tidyverse) # for data wrangling etc
library(rstanarm) # for fitting models in STAN
library(cmdstanr) # for cmdstan
library(brms) # for fitting models in STAN
library(standist) # for exploring distributions
library(HDInterval) # for HPD intervals
library(posterior) # for posterior draws
library(coda) # for diagnostics
library(bayesplot) # for diagnostics
library(ggmcmc) # for MCMC diagnostics
library(DHARMa) # for residual diagnostics
library(rstan) # for interfacing with STAN
library(emmeans) # for marginal means etc
library(broom) # for tidying outputs
library(tidybayes) # for more tidying outputs
library(ggeffects) # for partial plots
library(broom.mixed) # for summarising models
library(ggeffects) # for partial effects plots
library(bayestestR) # for ROPE
library(see) # for some plots
library(easystats) # for the easystats ecosystem
library(ggridges) # for ridge plots
library(patchwork) # for multiple plots
library(modelsummary) # for data and model summaries
theme_set(theme_grey()) # put the default ggplot theme back
source("helperFunctions.R")

2 Scenario

An ecologist studying a rocky shore at Phillip Island, in southeastern Australia, was interested in how clumps of intertidal mussels are maintained (Quinn 1988). In particular, he wanted to know how densities of adult mussels affected recruitment of young individuals from the plankton. As with most marine invertebrates, recruitment is highly patchy in time, so he expected to find seasonal variation, and the interaction between season and density - whether effects of adult mussel density vary across seasons - was the aspect of most interest.

The data were collected from four seasons, and with two densities of adult mussels. The experiment consisted of clumps of adult mussels attached to the rocks. These clumps were then brought back to the laboratory, and the number of baby mussels recorded. There were 3-6 replicate clumps for each density and season combination.

Table 1: Format of the quinn.csv data file
SEASON DENSITY RECRUITS SQRTRECRUITS GROUP
Spring Low 15 3.87 SpringLow
.. .. .. .. ..
Spring High 11 3.32 SpringHigh
.. .. .. .. ..
Summer Low 21 4.58 SummerLow
.. .. .. .. ..
Summer High 34 5.83 SummerHigh
.. .. .. .. ..
Autumn Low 14 3.74 AutumnLow
.. .. .. .. ..
Table 2: Description of the variables in the quinn data file
SEASON Categorical listing of Season in which mussel clumps were collected ­ independent variable
DENSITY Categorical listing of the density of mussels within mussel clump ­ independent variable
RECRUITS The number of mussel recruits ­ response variable
SQRTRECRUITS Square root transformation of RECRUITS - needed to meet the test assumptions
GROUPS Categorical listing of Season/Density combinations - used for checking ANOVA assumptions
Figure 1: Mussel

3 Read in the data

quinn <- read_csv("../data/quinn.csv", trim_ws = TRUE)
Rows: 42 Columns: 5
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (3): SEASON, DENSITY, GROUP
dbl (2): RECRUITS, SQRTRECRUITS

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
glimpse(quinn)
Rows: 42
Columns: 5
$ SEASON       <chr> "Spring", "Spring", "Spring", "Spring", "Spring", "Spring…
$ DENSITY      <chr> "Low", "Low", "Low", "Low", "Low", "High", "High", "High"…
$ RECRUITS     <dbl> 15, 10, 13, 13, 5, 11, 10, 15, 10, 13, 1, 21, 31, 21, 18,…
$ SQRTRECRUITS <dbl> 3.872983, 3.162278, 3.605551, 3.605551, 2.236068, 3.31662…
$ GROUP        <chr> "SpringLow", "SpringLow", "SpringLow", "SpringLow", "Spri…
## Explore the first 6 rows of the data
head(quinn)
# A tibble: 6 × 5
  SEASON DENSITY RECRUITS SQRTRECRUITS GROUP     
  <chr>  <chr>      <dbl>        <dbl> <chr>     
1 Spring Low           15         3.87 SpringLow 
2 Spring Low           10         3.16 SpringLow 
3 Spring Low           13         3.61 SpringLow 
4 Spring Low           13         3.61 SpringLow 
5 Spring Low            5         2.24 SpringLow 
6 Spring High          11         3.32 SpringHigh
str(quinn)
spc_tbl_ [42 × 5] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
 $ SEASON      : chr [1:42] "Spring" "Spring" "Spring" "Spring" ...
 $ DENSITY     : chr [1:42] "Low" "Low" "Low" "Low" ...
 $ RECRUITS    : num [1:42] 15 10 13 13 5 11 10 15 10 13 ...
 $ SQRTRECRUITS: num [1:42] 3.87 3.16 3.61 3.61 2.24 ...
 $ GROUP       : chr [1:42] "SpringLow" "SpringLow" "SpringLow" "SpringLow" ...
 - attr(*, "spec")=
  .. cols(
  ..   SEASON = col_character(),
  ..   DENSITY = col_character(),
  ..   RECRUITS = col_double(),
  ..   SQRTRECRUITS = col_double(),
  ..   GROUP = col_character()
  .. )
 - attr(*, "problems")=<externalptr> 
quinn |> datawizard::data_codebook()
quinn (42 rows and 5 variables, 5 shown)

ID | Name         | Type      | Missings |     Values |          N
---+--------------+-----------+----------+------------+-----------
1  | SEASON       | character | 0 (0.0%) |     Autumn | 10 (23.8%)
   |              |           |          |     Spring | 11 (26.2%)
   |              |           |          |     Summer | 12 (28.6%)
   |              |           |          |     Winter |  9 (21.4%)
---+--------------+-----------+----------+------------+-----------
2  | DENSITY      | character | 0 (0.0%) |       High | 24 (57.1%)
   |              |           |          |        Low | 18 (42.9%)
---+--------------+-----------+----------+------------+-----------
3  | RECRUITS     | numeric   | 0 (0.0%) |    [0, 69] |         42
---+--------------+-----------+----------+------------+-----------
4  | SQRTRECRUITS | numeric   | 0 (0.0%) |  [0, 8.31] |         42
---+--------------+-----------+----------+------------+-----------
5  | GROUP        | character | 0 (0.0%) | AutumnHigh |  6 (14.3%)
   |              |           |          |  AutumnLow |  4 ( 9.5%)
   |              |           |          | SpringHigh |  6 (14.3%)
   |              |           |          |  SpringLow |  5 (11.9%)
   |              |           |          | SummerHigh |  6 (14.3%)
   |              |           |          |  SummerLow |  6 (14.3%)
   |              |           |          | WinterHigh |  6 (14.3%)
   |              |           |          |  WinterLow |  3 ( 7.1%)
------------------------------------------------------------------
quinn |> modelsummary::datasummary_skim()
Unique Missing Pct. Mean SD Min Median Max Histogram
RECRUITS 25 0 18.3 15.7 0.0 13.5 69.0
SQRTRECRUITS 25 0 3.9 1.9 0.0 3.7 8.3
N %
SEASON Autumn 10 23.8
Spring 11 26.2
Summer 12 28.6
Winter 9 21.4
DENSITY High 24 57.1
Low 18 42.9
GROUP AutumnHigh 6 14.3
AutumnLow 4 9.5
SpringHigh 6 14.3
SpringLow 5 11.9
SummerHigh 6 14.3
SummerLow 6 14.3
WinterHigh 6 14.3
WinterLow 3 7.1
quinn |>
  dplyr::select(-SQRTRECRUITS) |>
  modelsummary::datasummary_skim(
    type = "numeric",
    by = c("SEASON", "DENSITY")
  )
SEASON DENSITY Unique Missing Pct. Mean SD Min Median Max Histogram
RECRUITS Spring Low 4 0 11.2 3.9 5.0 13.0 15.0
Spring High 5 0 10.0 4.8 1.0 10.5 15.0
Summer Low 5 0 22.0 6.1 14.0 21.0 31.0
Summer High 6 0 48.2 15.0 28.0 51.5 69.0
Autumn Low 4 0 18.2 3.1 14.0 19.0 21.0
Autumn High 5 0 19.7 11.9 4.0 17.5 36.0
Winter Low 2 0 2.7 4.6 0.0 0.0 8.0
Winter High 5 0 5.7 3.3 1.0 5.0 10.0

4 Exploratory data analysis

Model formula: \[ \begin{align} y_i &\sim{} \mathcal{NB}(\lambda_i, \theta)\\ ln(\mu_i) &= \beta_0 + \sum_{j=1}^nT_{[i],j}.\beta_j\\ \beta_0 &\sim{} \mathcal{N}(2.4, 1.5)\\ \beta_{[1,2,3]} &\sim{} \mathcal{N}(0, 1)\\ \end{align} \]

where \(\beta_{0}\) is the y-intercept (mean of the first group), \(\beta_{[1,2,3]}\) are the vector of effects parameters (contrasting each group mean to that of the first group and \(T{[i],j}\) represents a \(i\) by \(j\) model matrix is a model matrix representing the season, density and their interaction on mussel recruitment.

quinn <- quinn |>
  mutate(
    fSEASON = factor(SEASON,
      levels = c("Spring", "Summer", "Autumn", "Winter")
    ),
    DENSITY = factor(DENSITY)
  )

5 Exploratory data analysis

The exploratory data analyses that we performed in the frequentist instalment of this example are equally valid here. That is, boxplots and/or violin plots for each population (substrate type).

quinn |> head()
# A tibble: 6 × 6
  SEASON DENSITY RECRUITS SQRTRECRUITS GROUP      fSEASON
  <chr>  <fct>      <dbl>        <dbl> <chr>      <fct>  
1 Spring Low           15         3.87 SpringLow  Spring 
2 Spring Low           10         3.16 SpringLow  Spring 
3 Spring Low           13         3.61 SpringLow  Spring 
4 Spring Low           13         3.61 SpringLow  Spring 
5 Spring Low            5         2.24 SpringLow  Spring 
6 Spring High          11         3.32 SpringHigh Spring 
quinn |> ggplot(aes(y = RECRUITS, x = fSEASON, fill = DENSITY)) +
  geom_boxplot()

Conclusions:

  • there is clearly a relationship between mean and variance (as would be expected with the a Poisson
  • evidently there are numerous zeros in the Winter/Low group # Fit the model

In rstanarm, the default priors are designed to be weakly informative. They are chosen to provide moderate regularisation (to help prevent over-fitting) and help stabilise the computations.

quinn.rstanarmP <- stan_glm(RECRUITS ~ fSEASON * DENSITY,
  data = quinn,
  family = poisson(link = "log"),
  refresh = 0,
  chains = 3, iter = 5000, thin = 5, warmup = 2000
)
quinn.rstanarmP |> prior_summary()
Priors for model 'quinn.rstanarmP' 
------
Intercept (after predictors centered)
 ~ normal(location = 0, scale = 2.5)

Coefficients
  Specified prior:
    ~ normal(location = [0,0,0,...], scale = [2.5,2.5,2.5,...])
  Adjusted prior:
    ~ normal(location = [0,0,0,...], scale = [5.47,5.80,6.02,...])
------
See help('prior_summary.stanreg') for more details

This tells us:

  • for the intercept, when the family is Poisson, it is using a normal prior with a mean of 0 and a standard deviation of 2.5. The 2.5 is used for all intercepts. It is often scaled, but only if it is larger than 2.5 is the scaled version kept.

  • for the coefficients (in this case, the individual effects), the default prior is a normal prior centred around 0 with a standard deviations of 5.47, 5/8, 6.02 etc. This is then adjusted for the scale of the data by dividing the 2.5 by the standard deviation of the numerical dummy variables for the predictor (then rounded).

2.5 / apply(model.matrix(~ fSEASON * DENSITY, quinn)[, -1], 2, sd)
           fSEASONSummer            fSEASONAutumn            fSEASONWinter 
                5.467708                 5.799380                 6.019749 
              DENSITYLow fSEASONSummer:DENSITYLow fSEASONAutumn:DENSITYLow 
                4.991312                 7.058781                 8.414625 
fSEASONWinter:DENSITYLow 
                9.590995 
  • there is no auxiliary prior as we are employing a Poisson distribution.
quinn.rstanarm1 <- stan_glm(RECRUITS ~ fSEASON * DENSITY,
  data = quinn,
  family = poisson(link = "log"),
  prior_PD = TRUE,
  refresh = 0,
  chains = 3, iter = 5000, thin = 5, warmup = 2000
)
quinn.rstanarm1 |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

quinn.rstanarm1 |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE) |>
  wrap_plots() &
  scale_y_log10()
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Warning in scale_y_log10(): log-10 transformation introduced infinite values.

## although, since there are zeros...
quinn.rstanarm1 |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE, jitter = FALSE) |>
  wrap_plots() &
  scale_y_continuous(trans = scales::pseudo_log_trans())
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.

quinn.rstanarm1 |>
  ggemmeans(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE) |>
  plot(show_data = TRUE) |>
  wrap_plots() &
  scale_y_log10()
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Warning in scale_y_log10(): log-10 transformation introduced infinite values.

## although, since there are zeros...
quinn.rstanarm1 |>
  ggemmeans(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE, jitter = FALSE) |>
  wrap_plots() &
  scale_y_continuous(trans = scales::pseudo_log_trans())
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.

Conclusions:

  • we see that the range of predictions is fairly wide and the predicted means could range from 0 to very large (perhaps too large).

The following link provides some guidance about defining priors. [https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations]

When defining our own priors, we typically do not want them to be scaled.

If we wanted to define our own priors that were less vague, yet still not likely to bias the outcomes, we could try the following priors (mainly plucked out of thin air):

  • \(\beta_0\): normal centred at 2.3 with a standard deviation of 5
  • \(\beta_1\): normal centred at 0 with a standard deviation of 2

Remember the above are applied on the link scale.

I will also overlay the raw data for comparison.

quinn |>
  group_by(fSEASON, DENSITY) |>
  summarise(
    Mean = log(mean(RECRUITS)),
    SD = log(sd(RECRUITS))
  )
`summarise()` has grouped output by 'fSEASON'. You can override using the
`.groups` argument.
# A tibble: 8 × 4
# Groups:   fSEASON [4]
  fSEASON DENSITY  Mean    SD
  <fct>   <fct>   <dbl> <dbl>
1 Spring  High    2.30   1.57
2 Spring  Low     2.42   1.36
3 Summer  High    3.87   2.71
4 Summer  Low     3.09   1.81
5 Autumn  High    2.98   2.48
6 Autumn  Low     2.90   1.13
7 Winter  High    1.73   1.20
8 Winter  Low     0.981  1.53
log(sd(quinn$RECRUITS)) /
  apply(model.matrix(~ fSEASON * DENSITY, data = quinn), 2, sd)
             (Intercept)            fSEASONSummer            fSEASONAutumn 
                     Inf                 6.024998                 6.390475 
           fSEASONWinter               DENSITYLow fSEASONSummer:DENSITYLow 
                6.633304                 5.500045                 7.778238 
fSEASONAutumn:DENSITYLow fSEASONWinter:DENSITYLow 
                9.272276                10.568545 
quinn.rstanarm2 <- stan_glm(RECRUITS ~ fSEASON * DENSITY,
  data = quinn,
  family = poisson(link = "log"),
  prior_intercept = normal(2.3, 2, autoscale = FALSE),
  prior = normal(0, 10, autoscale = FALSE),
  prior_PD = TRUE,
  refresh = 0,
  chains = 3, iter = 5000, thin = 5, warmup = 2000
)
quinn.rstanarm2 |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

quinn.rstanarm2 |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE) |>
  wrap_plots() &
  scale_y_log10()
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Warning in scale_y_log10(): log-10 transformation introduced infinite values.

## although, since there are zeros...
quinn.rstanarm2 |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE, jitter = FALSE) |>
  wrap_plots() &
  scale_y_continuous(trans = scales::pseudo_log_trans())
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.

quinn.rstanarm2 |>
  ggemmeans(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE) |>
  plot(show_data = TRUE) |>
  wrap_plots() &
  scale_y_log10()
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Warning in scale_y_log10(): log-10 transformation introduced infinite values.

## although, since there are zeros...
quinn.rstanarm2 |>
  ggemmeans(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE, jitter = FALSE) |>
  wrap_plots() &
  scale_y_continuous(trans = scales::pseudo_log_trans())
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.

Now lets refit, conditioning on the data.

quinn.rstanarm3 <- quinn.rstanarm2 |> update(prior_PD = FALSE)
quinn.rstanarm3 |> posterior_vs_prior(
  color_by = "vs", group_by = TRUE,
  facet_args = list(scales = "free_y")
)

Drawing from prior...

Conclusions:

  • in each case, the prior is substantially wider than the posterior, suggesting that the posterior is not biased towards the prior.
quinn.rstanarm3 |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

quinn.rstanarm3 |>
  ggemmeans(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

In brms, the default priors are designed to be weakly informative. They are chosen to provide moderate regularisation (to help prevent over-fitting) and help stabilise the computations.

Unlike rstanarm, brms models must be compiled before they start sampling. For most models, the compilation of the stan code takes around 45 seconds.

quinn.form <- bf(RECRUITS ~ fSEASON * DENSITY, family = poisson(link = "log"))
get_prior(quinn.form, data = quinn)
                  prior     class                     coef group resp dpar nlpar lb ub       source
                 (flat)         b                                                           default
                 (flat)         b               DENSITYLow                             (vectorized)
                 (flat)         b            fSEASONAutumn                             (vectorized)
                 (flat)         b fSEASONAutumn:DENSITYLow                             (vectorized)
                 (flat)         b            fSEASONSummer                             (vectorized)
                 (flat)         b fSEASONSummer:DENSITYLow                             (vectorized)
                 (flat)         b            fSEASONWinter                             (vectorized)
                 (flat)         b fSEASONWinter:DENSITYLow                             (vectorized)
 student_t(3, 2.6, 2.5) Intercept                                                           default

Remember that the priors are applied on the link (in this case, log) scale.

quinn |>
  group_by(fSEASON, DENSITY) |>
  summarise(
    Mean = mean(RECRUITS),
    Median = median(RECRUITS),
    MAD = mad(RECRUITS),
    SD = sd(RECRUITS)
  ) |>
  mutate(
    log(Mean),
    log(Median),
    log(MAD),
    log(SD)
  )
`summarise()` has grouped output by 'fSEASON'. You can override using the
`.groups` argument.
# A tibble: 8 × 10
# Groups:   fSEASON [4]
  fSEASON DENSITY  Mean Median   MAD    SD `log(Mean)` `log(Median)` `log(MAD)`
  <fct>   <fct>   <dbl>  <dbl> <dbl> <dbl>       <dbl>         <dbl>      <dbl>
1 Spring  High    10      10.5  2.22  4.82       2.30           2.35      0.799
2 Spring  Low     11.2    13    2.97  3.90       2.42           2.56      1.09 
3 Summer  High    48.2    51.5 15.6  15.0        3.87           3.94      2.75 
4 Summer  Low     22      21    6.67  6.13       3.09           3.04      1.90 
5 Autumn  High    19.7    17.5 12.6  11.9        2.98           2.86      2.53 
6 Autumn  Low     18.2    19    2.22  3.10       2.90           2.94      0.799
7 Winter  High     5.67    5    3.71  3.33       1.73           1.61      1.31 
8 Winter  Low      2.67    0    0     4.62       0.981       -Inf      -Inf    
# ℹ 1 more variable: `log(SD)` <dbl>
  • \(\beta_0\): normal centred at 2.3 with a standard deviation of 1.5
  • \(\beta_1\): normal centred at 0 with a standard deviation of 1
priors <- prior(normal(2.4, 1.5), class = "Intercept") +
  prior(normal(0, 1), class = "b")
quinn.brm2 <- brm(quinn.form,
  data = quinn,
  prior = priors,
  sample_prior = "only",
  refresh = 0,
  chains = 3,
  iter = 5000,
  thin = 5,
  warmup = 2500,
  backend = "rstan"
)
Compiling Stan program...
Start sampling
quinn.brm2 |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

quinn.brm2 |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE) +
  scale_y_log10()
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Warning in scale_y_log10(): log-10 transformation introduced infinite values.

## Or since there are zeros
quinn.brm2 |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE) +
  scale_y_continuous(trans = scales::pseudo_log_trans())
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.

quinn.brm2 |>
  ggemmeans(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

quinn.brm2 |>
  ggemmeans(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE) +
  scale_y_log10()
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Warning in scale_y_log10(): log-10 transformation introduced infinite values.

## Or since there are zeros
quinn.brm2 |>
  ggemmeans(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE) +
  scale_y_continuous(trans = scales::pseudo_log_trans())
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.

quinn.brm2 |>
  conditional_effects("fSEASON:DENSITY") |>
  plot(points = TRUE)

quinn.brm2 |>
  conditional_effects("fSEASON:DENSITY") |>
  plot(points = TRUE) |>
  wrap_plots() &
  scale_y_continuous(trans = scales::pseudo_log_trans())

quinn.brmP <- quinn.brm2 |> update(sample_prior = "yes", refresh = 0)
The desired updates require recompiling the model
Compiling Stan program...
Start sampling
quinn.brmP |> get_variables()
 [1] "b_Intercept"                "b_fSEASONSummer"           
 [3] "b_fSEASONAutumn"            "b_fSEASONWinter"           
 [5] "b_DENSITYLow"               "b_fSEASONSummer:DENSITYLow"
 [7] "b_fSEASONAutumn:DENSITYLow" "b_fSEASONWinter:DENSITYLow"
 [9] "Intercept"                  "prior_Intercept"           
[11] "prior_b"                    "lprior"                    
[13] "lp__"                       "accept_stat__"             
[15] "stepsize__"                 "treedepth__"               
[17] "n_leapfrog__"               "divergent__"               
[19] "energy__"                  
quinn.brmP |>
  hypothesis("fSEASONSummer<0") |>
  plot()

quinn.brmP |>
  hypothesis("DENSITYLow<0") |>
  plot()

quinn.brmP |>
  hypothesis("fSEASONSummer:DENSITYLow<0") |>
  plot()

quinn.brmP |> SUYR_prior_and_posterior()

quinn.brmP |>
  posterior_samples() |>
  dplyr::select(-`lp__`) |>
  pivot_longer(everything(), names_to = "key") |>
  mutate(
    Type = ifelse(str_detect(key, "prior"), "Prior", "b"),
    Class = ifelse(str_detect(key, "Intercept"), "Intercept",
      ifelse(str_detect(key, "b"), "b", "sigma")
    ),
    Par = str_replace(key, "b_", "")
  ) |>
  ggplot(aes(x = Type, y = value, color = Par)) +
  stat_pointinterval(position = position_dodge()) +
  facet_wrap(~Class, scales = "free")
Warning: Method 'posterior_samples' is deprecated. Please see ?as_draws for
recommended alternatives.

quinn.brmP |> standata()
$N
[1] 42

$Y
 [1] 15 10 13 13  5 11 10 15 10 13  1 21 31 21 18 14 27 34 49 69 55 28 54 14 18
[26] 20 21  4 22 30 36 13 13  8  0  0 10  1  5  9  4  5

$K
[1] 8

$Kc
[1] 7

$X
   Intercept fSEASONSummer fSEASONAutumn fSEASONWinter DENSITYLow
1          1             0             0             0          1
2          1             0             0             0          1
3          1             0             0             0          1
4          1             0             0             0          1
5          1             0             0             0          1
6          1             0             0             0          0
7          1             0             0             0          0
8          1             0             0             0          0
9          1             0             0             0          0
10         1             0             0             0          0
11         1             0             0             0          0
12         1             1             0             0          1
13         1             1             0             0          1
14         1             1             0             0          1
15         1             1             0             0          1
16         1             1             0             0          1
17         1             1             0             0          1
18         1             1             0             0          0
19         1             1             0             0          0
20         1             1             0             0          0
21         1             1             0             0          0
22         1             1             0             0          0
23         1             1             0             0          0
24         1             0             1             0          1
25         1             0             1             0          1
26         1             0             1             0          1
27         1             0             1             0          1
28         1             0             1             0          0
29         1             0             1             0          0
30         1             0             1             0          0
31         1             0             1             0          0
32         1             0             1             0          0
33         1             0             1             0          0
34         1             0             0             1          1
35         1             0             0             1          1
36         1             0             0             1          1
37         1             0             0             1          0
38         1             0             0             1          0
39         1             0             0             1          0
40         1             0             0             1          0
41         1             0             0             1          0
42         1             0             0             1          0
   fSEASONSummer:DENSITYLow fSEASONAutumn:DENSITYLow fSEASONWinter:DENSITYLow
1                         0                        0                        0
2                         0                        0                        0
3                         0                        0                        0
4                         0                        0                        0
5                         0                        0                        0
6                         0                        0                        0
7                         0                        0                        0
8                         0                        0                        0
9                         0                        0                        0
10                        0                        0                        0
11                        0                        0                        0
12                        1                        0                        0
13                        1                        0                        0
14                        1                        0                        0
15                        1                        0                        0
16                        1                        0                        0
17                        1                        0                        0
18                        0                        0                        0
19                        0                        0                        0
20                        0                        0                        0
21                        0                        0                        0
22                        0                        0                        0
23                        0                        0                        0
24                        0                        1                        0
25                        0                        1                        0
26                        0                        1                        0
27                        0                        1                        0
28                        0                        0                        0
29                        0                        0                        0
30                        0                        0                        0
31                        0                        0                        0
32                        0                        0                        0
33                        0                        0                        0
34                        0                        0                        1
35                        0                        0                        1
36                        0                        0                        1
37                        0                        0                        0
38                        0                        0                        0
39                        0                        0                        0
40                        0                        0                        0
41                        0                        0                        0
42                        0                        0                        0
attr(,"assign")
[1] 0 1 1 1 2 3 3 3
attr(,"contrasts")
attr(,"contrasts")$fSEASON
       Summer Autumn Winter
Spring      0      0      0
Summer      1      0      0
Autumn      0      1      0
Winter      0      0      1

attr(,"contrasts")$DENSITY
     Low
High   0
Low    1


$prior_only
[1] 0

attr(,"class")
[1] "standata" "list"    
quinn.brmP |> stancode()
// generated with brms 2.22.0
functions {
}
data {
  int<lower=1> N;  // total number of observations
  array[N] int Y;  // response variable
  int<lower=1> K;  // number of population-level effects
  matrix[N, K] X;  // population-level design matrix
  int<lower=1> Kc;  // number of population-level effects after centering
  int prior_only;  // should the likelihood be ignored?
}
transformed data {
  matrix[N, Kc] Xc;  // centered version of X without an intercept
  vector[Kc] means_X;  // column means of X before centering
  for (i in 2:K) {
    means_X[i - 1] = mean(X[, i]);
    Xc[, i - 1] = X[, i] - means_X[i - 1];
  }
}
parameters {
  vector[Kc] b;  // regression coefficients
  real Intercept;  // temporary intercept for centered predictors
}
transformed parameters {
  real lprior = 0;  // prior contributions to the log posterior
  lprior += normal_lpdf(b | 0, 1);
  lprior += normal_lpdf(Intercept | 2.4, 1.5);
}
model {
  // likelihood including constants
  if (!prior_only) {
    target += poisson_log_glm_lpmf(Y | Xc, Intercept, b);
  }
  // priors including constants
  target += lprior;
}
generated quantities {
  // actual population-level intercept
  real b_Intercept = Intercept - dot_product(means_X, b);
  // additionally sample draws from priors
  real prior_b = normal_rng(0,1);
  real prior_Intercept = normal_rng(2.4,1.5);
}

6 MCMC sampling diagnostics

The bayesplot package offers a range of MCMC diagnostics as well as Posterior Probability Checks (PPC), all of which have a convenient plot() interface. Lets start with the MCMC diagnostics.

available_mcmc()
bayesplot MCMC module:
  mcmc_acf
  mcmc_acf_bar
  mcmc_areas
  mcmc_areas_ridges
  mcmc_combo
  mcmc_dens
  mcmc_dens_chains
  mcmc_dens_overlay
  mcmc_hex
  mcmc_hist
  mcmc_hist_by_chain
  mcmc_intervals
  mcmc_neff
  mcmc_neff_hist
  mcmc_nuts_acceptance
  mcmc_nuts_divergence
  mcmc_nuts_energy
  mcmc_nuts_stepsize
  mcmc_nuts_treedepth
  mcmc_pairs
  mcmc_parcoord
  mcmc_rank_ecdf
  mcmc_rank_hist
  mcmc_rank_overlay
  mcmc_recover_hist
  mcmc_recover_intervals
  mcmc_recover_scatter
  mcmc_rhat
  mcmc_rhat_hist
  mcmc_scatter
  mcmc_trace
  mcmc_trace_highlight
  mcmc_violin

Of these, we will focus on:

  • mcmc_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
plot(quinn.rstanarm3, plotfun = "mcmc_trace")

The chains appear well mixed and very similar

  • acf (auto-correlation function): plots the auto-correlation between successive MCMC sample lags for each parameter and each chain
plot(quinn.rstanarm3, "acf_bar")

There is no evidence of auto-correlation in the MCMC samples

  • Rhat: Rhat is a measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
plot(quinn.rstanarm3, "rhat_hist")
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

All Rhat values are below 1.05, suggesting the chains have converged.

  • neff (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

    If the ratios are low, tightening the priors may help.

plot(quinn.rstanarm3, "neff_hist")
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Ratios all very high.

More diagnostics
plot(quinn.rstanarm3, "combo")

plot(quinn.rstanarm3, "violin")

The rstan package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.

Of these, we will focus on:

  • stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
stan_trace(quinn.rstanarm3)

The chains appear well mixed and very similar

  • stan_acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
stan_ac(quinn.rstanarm3)

There is no evidence of auto-correlation in the MCMC samples

  • stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
stan_rhat(quinn.rstanarm3)
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

All Rhat values are below 1.05, suggesting the chains have converged.

  • stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

    If the ratios are low, tightening the priors may help.

stan_ess(quinn.rstanarm3)
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Ratios all very high.

stan_dens(quinn.rstanarm3, separate_chains = TRUE)

The ggmean package also has a set of MCMC diagnostic functions. Lets start with the MCMC diagnostics.

Of these, we will focus on:

  • ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
quinn.ggs <- ggs(quinn.rstanarm3, burnin = FALSE, inc_warmup = FALSE)
ggs_traceplot(quinn.ggs)

The chains appear well mixed and very similar

  • gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
ggs_autocorrelation(quinn.ggs)

There is no evidence of auto-correlation in the MCMC samples

  • stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
ggs_Rhat(quinn.ggs)

All Rhat values are below 1.05, suggesting the chains have converged.

  • stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

    If the ratios are low, tightening the priors may help.

ggs_effective(quinn.ggs)
Warning: Returning more (or less) than 1 row per `summarise()` group was deprecated in
dplyr 1.1.0.
ℹ Please use `reframe()` instead.
ℹ When switching from `summarise()` to `reframe()`, remember that `reframe()`
  always returns an ungrouped data frame and adjust accordingly.
ℹ The deprecated feature was likely used in the ggmcmc package.
  Please report the issue at <https://github.com/xfim/ggmcmc/issues/>.

Ratios all very high.

More diagnostics
ggs_crosscorrelation(quinn.ggs)

ggs_grb(quinn.ggs)

The brms package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.

Of these, we will focus on:

  • stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
quinn.brmP$fit |> stan_trace()
'pars' not specified. Showing first 10 parameters by default.

quinn.brmP$fit |> stan_trace(inc_warmup = TRUE)
'pars' not specified. Showing first 10 parameters by default.

The chains appear well mixed and very similar

  • stan_acf (auto-correlation function): plots the auto-correlation between successive MCMC sample lags for each parameter and each chain
quinn.brmP$fit |> stan_ac()
'pars' not specified. Showing first 10 parameters by default.

There is no evidence of auto-correlation in the MCMC samples

  • stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
quinn.brmP$fit |> stan_rhat()
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

All Rhat values are below 1.05, suggesting the chains have converged.

  • stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).

    If the ratios are low, tightening the priors may help.

quinn.brmP$fit |> stan_ess()
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Ratios all very high.

quinn.brmP$fit |> stan_dens(separate_chains = TRUE)
'pars' not specified. Showing first 10 parameters by default.

7 Model validation

Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.

available_ppc()
bayesplot PPC module:
  ppc_bars
  ppc_bars_grouped
  ppc_boxplot
  ppc_dens
  ppc_dens_overlay
  ppc_dens_overlay_grouped
  ppc_ecdf_overlay
  ppc_ecdf_overlay_grouped
  ppc_error_binned
  ppc_error_hist
  ppc_error_hist_grouped
  ppc_error_scatter
  ppc_error_scatter_avg
  ppc_error_scatter_avg_grouped
  ppc_error_scatter_avg_vs_x
  ppc_freqpoly
  ppc_freqpoly_grouped
  ppc_hist
  ppc_intervals
  ppc_intervals_grouped
  ppc_km_overlay
  ppc_km_overlay_grouped
  ppc_loo_intervals
  ppc_loo_pit_ecdf
  ppc_loo_pit_overlay
  ppc_loo_pit_qq
  ppc_loo_ribbon
  ppc_pit_ecdf
  ppc_pit_ecdf_grouped
  ppc_ribbon
  ppc_ribbon_grouped
  ppc_rootogram
  ppc_scatter
  ppc_scatter_avg
  ppc_scatter_avg_grouped
  ppc_stat
  ppc_stat_2d
  ppc_stat_freqpoly
  ppc_stat_freqpoly_grouped
  ppc_stat_grouped
  ppc_violin_grouped
  • dens_overlay: plots the density distribution of the observed data (black line) overlayed on top of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.
pp_check(quinn.rstanarm3, plotfun = "dens_overlay")

The model draws are broadly similar to the observed data.

  • error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. There is some pattern remaining in these residuals.
pp_check(quinn.rstanarm3, plotfun = "error_scatter_avg")

The predictive error seems to be related to the predictor - the model performs poorest at higher recruitments

  • error_scatter_avg_vs_x: this is similar to a regular residual plot and as such should be interpreted as such.
pp_check(quinn.rstanarm3, x = as.numeric(quinn$fSEASON), plotfun = "error_scatter_avg_vs_x")

pp_check(quinn.rstanarm3, x = as.numeric(quinn$DENSITY), plotfun = "error_scatter_avg_vs_x")

  • intervals: plots the observed data overlayed on top of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.
pp_check(quinn.rstanarm3, x = as.numeric(quinn$fSEASON), plotfun = "intervals")

The modelled predictions seem to do a reasonable job of representing the observations.

The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.

# library(shinystan)
# launch_shinystan(quinn.rstanarm3)

DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components yourself, we can still obtain the simulated residuals from the fitted stan model.

We need to supply:

  • simulated (predicted) responses associated with each observation.
  • observed values
  • fitted (predicted) responses (averaged) associated with each observation
preds <- posterior_predict(quinn.rstanarm3, nsamples = 250, summary = FALSE)
quinn.resids <- createDHARMa(
  simulatedResponse = t(preds),
  observedResponse = quinn$RECRUITS,
  fittedPredictedResponse = apply(preds, 2, median),
  integerResponse = TRUE
)
plot(quinn.resids)

Conclusions:

  • the simulated residuals suggest there might be an issue of dispersion.
  • it might be worth exploring either zero-inflation, a negative binomial model, or including a observation-level random effect.

Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.

available_ppc()
bayesplot PPC module:
  ppc_bars
  ppc_bars_grouped
  ppc_boxplot
  ppc_dens
  ppc_dens_overlay
  ppc_dens_overlay_grouped
  ppc_ecdf_overlay
  ppc_ecdf_overlay_grouped
  ppc_error_binned
  ppc_error_hist
  ppc_error_hist_grouped
  ppc_error_scatter
  ppc_error_scatter_avg
  ppc_error_scatter_avg_grouped
  ppc_error_scatter_avg_vs_x
  ppc_freqpoly
  ppc_freqpoly_grouped
  ppc_hist
  ppc_intervals
  ppc_intervals_grouped
  ppc_km_overlay
  ppc_km_overlay_grouped
  ppc_loo_intervals
  ppc_loo_pit_ecdf
  ppc_loo_pit_overlay
  ppc_loo_pit_qq
  ppc_loo_ribbon
  ppc_pit_ecdf
  ppc_pit_ecdf_grouped
  ppc_ribbon
  ppc_ribbon_grouped
  ppc_rootogram
  ppc_scatter
  ppc_scatter_avg
  ppc_scatter_avg_grouped
  ppc_stat
  ppc_stat_2d
  ppc_stat_freqpoly
  ppc_stat_freqpoly_grouped
  ppc_stat_grouped
  ppc_violin_grouped
  • dens_overlay: plots the density distribution of the observed data (black line) overlayed on top of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.
quinn.brmP |> pp_check(type = "dens_overlay", ndraws = 100)

The model draws appear to represent the shape of the observed data reasonably well

  • error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. There is some pattern remaining in these residuals.
quinn.brmP |> pp_check(type = "error_scatter_avg")
Using all posterior draws for ppc type 'error_scatter_avg' by default.

The predictive error seems to be related to the predictor - the model performs poorest at higher recruitments.

  • intervals: plots the observed data overlayed on top of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.
quinn.brmP |> pp_check(type = "intervals")
Using all posterior draws for ppc type 'intervals' by default.

## quinn.brmP |> pp_check(group='DENSITY', type='intervals')

Whilst the modelled predictions do a reasonable job of representing the observed data, the observed data do appear to be more varied than the model is representing.

The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.

# library(shinystan)
# launch_shinystan(quinn.brmP)

DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components yourself, we can still obtain the simulated residuals from the fitted stan model.

We need to supply:

  • simulated (predicted) responses associated with each observation.
  • observed values
  • fitted (predicted) responses (averaged) associated with each observation
preds <- quinn.brmP |> posterior_predict(nsamples = 250, summary = FALSE)
Warning: Argument 'nsamples' is deprecated. Please use argument 'ndraws'
instead.
quinn.resids <- createDHARMa(
  simulatedResponse = t(preds),
  observedResponse = quinn$RECRUITS,
  fittedPredictedResponse = apply(preds, 2, median),
  integerResponse = TRUE
)
quinn.resids |> plot()

quinn.resids |> testDispersion()


    DHARMa nonparametric dispersion test via sd of residuals fitted vs.
    simulated

data:  simulationOutput
dispersion = 2.6852, p-value = 0.008
alternative hypothesis: two.sided
quinn.resids |> testZeroInflation()


    DHARMa zero-inflation test via comparison to expected zeros with
    simulation under H0 = fitted model

data:  simulationOutput
ratioObsSim = 6.5789, p-value = 0.08
alternative hypothesis: two.sided
quinn.resids <- make_brms_dharma_res(quinn.brmP, integerResponse = TRUE)
wrap_elements(~ testUniformity(quinn.resids)) +
  wrap_elements(~ plotResiduals(quinn.resids, form = factor(rep(1, nrow(quinn))))) +
  wrap_elements(~ plotResiduals(quinn.resids, quantreg = TRUE)) +
  wrap_elements(~ testDispersion(quinn.resids))

Conclusions:

  • the simulated residuals do suggest that there might be a dispersion issue
  • it might be worth exploring either zero-inflation, a negative binomial model, or including a observation-level random effect.

8 Explore negative binomial model

quinn.rstanarmNB <- stan_glm(RECRUITS ~ fSEASON * DENSITY,
  data = quinn,
  family = neg_binomial_2(link = "log"),
  prior_intercept = normal(2.3, 2, autoscale = FALSE),
  prior = normal(0, 10, autoscale = FALSE),
  prior_aux = rstanarm::exponential(rate = 1, autoscale = FALSE),
  prior_PD = FALSE,
  refresh = 0,
  chains = 3, iter = 5000, thin = 5, warmup = 2000
)
posterior_vs_prior(quinn.rstanarmNB,
  color_by = "vs", group_by = TRUE,
  facet_args = list(scales = "free_y")
)

Drawing from prior...

quinn.rstanarmNB |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

quinn.rstanarmNB |>
  ggemmeans(~ fSEASON + DENSITY, back.transform = TRUE) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

There seems to be a bug here. The expected values should be being back transformed to the response scale, however, they are clearly not. Notice that the expected values (and associated CI) are low and tiny respectively).

quinn.rstanarmNB |> plot("mcmc_trace")

quinn.rstanarmNB |> plot("mcmc_acf_bar")

quinn.rstanarmNB |> plot("mcmc_rhat_hist")
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

quinn.rstanarmNB |> plot("mcmc_neff_hist")
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

preds <- posterior_predict(quinn.rstanarmNB, nsamples = 250, summary = FALSE)
quinn.resids <- createDHARMa(
  simulatedResponse = t(preds),
  observedResponse = quinn$RECRUITS,
  fittedPredictedResponse = apply(preds, 2, median),
  integerResponse = TRUE
)
plot(quinn.resids)

quinn.resids |> testDispersion()


    DHARMa nonparametric dispersion test via sd of residuals fitted vs.
    simulated

data:  simulationOutput
dispersion = 0.27522, p-value = 0.03222
alternative hypothesis: two.sided

Now possibly under-dispersed..

(loo.P <- loo(quinn.rstanarmP))
Warning: Found 2 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 2 times to compute the ELPDs for the problematic observations directly.

Computed from 1800 by 42 log-likelihood matrix.

         Estimate   SE
elpd_loo   -171.0 15.7
p_loo        24.3  5.7
looic       342.0 31.3
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.8, 1.1]).

Pareto k diagnostic values:
                          Count Pct.    Min. ESS
(-Inf, 0.69]   (good)     40    95.2%   161     
   (0.69, 1]   (bad)       1     2.4%   <NA>    
    (1, Inf)   (very bad)  1     2.4%   <NA>    
See help('pareto-k-diagnostic') for details.
(loo.NB <- loo(quinn.rstanarmNB))
Warning: Found 1 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 1 times to compute the ELPDs for the problematic observations directly.

Computed from 1800 by 42 log-likelihood matrix.

         Estimate   SE
elpd_loo   -150.4  6.0
p_loo         9.3  3.5
looic       300.9 12.0
------
MCSE of elpd_loo is NA.
MCSE and ESS estimates assume MCMC draws (r_eff in [0.9, 1.1]).

Pareto k diagnostic values:
                          Count Pct.    Min. ESS
(-Inf, 0.69]   (good)     41    97.6%   622     
   (0.69, 1]   (bad)       0     0.0%   <NA>    
    (1, Inf)   (very bad)  1     2.4%   <NA>    
See help('pareto-k-diagnostic') for details.
loo_compare(loo.P, loo.NB)
                 elpd_diff se_diff
quinn.rstanarmNB   0.0       0.0  
quinn.rstanarmP  -20.5      11.9  
quinn.form <- bf(RECRUITS ~ fSEASON * DENSITY, family = negbinomial(link = "log"))
get_prior(quinn.form, data = quinn)
                  prior     class                     coef group resp dpar
                 (flat)         b                                         
                 (flat)         b               DENSITYLow                
                 (flat)         b            fSEASONAutumn                
                 (flat)         b fSEASONAutumn:DENSITYLow                
                 (flat)         b            fSEASONSummer                
                 (flat)         b fSEASONSummer:DENSITYLow                
                 (flat)         b            fSEASONWinter                
                 (flat)         b fSEASONWinter:DENSITYLow                
 student_t(3, 2.6, 2.5) Intercept                                         
    inv_gamma(0.4, 0.3)     shape                                         
 nlpar lb ub       source
                  default
             (vectorized)
             (vectorized)
             (vectorized)
             (vectorized)
             (vectorized)
             (vectorized)
             (vectorized)
                  default
        0         default
priors <- prior(normal(2.4, 1.5), class = "Intercept") +
  prior(normal(0, 1), class = "b") +
  prior(gamma(0.01, 0.01), class = "shape")
quinn.brmsNB <- brm(quinn.form,
  data = quinn,
  prior = priors,
  refresh = 0,
  chains = 3,
  iter = 5000,
  thin = 5,
  warmup = 2500,
  backend = "rstan"
)
Compiling Stan program...
Start sampling
preds <- posterior_predict(quinn.brmsNB, nsamples = 250, summary = FALSE)
Warning: Argument 'nsamples' is deprecated. Please use argument 'ndraws'
instead.
quinn.resids <- createDHARMa(
  simulatedResponse = t(preds),
  observedResponse = quinn$RECRUITS,
  fittedPredictedResponse = apply(preds, 2, median),
  integerResponse = TRUE
)
plot(quinn.resids)

quinn.resids <- make_brms_dharma_res(quinn.brmsNB, integerResponse = TRUE)
wrap_elements(~ testUniformity(quinn.resids)) +
  ## wrap_elements(~plotResiduals(quinn.resids, form = factor(rep(1, nrow(quinn))))) +
  wrap_elements(~ plotResiduals(quinn.resids, quantreg = TRUE)) +
  wrap_elements(~ testDispersion(quinn.resids))

9 Partial effects plots

quinn.rstanarmNB |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

quinn.rstanarmNB |>
  ggemmeans(~ fSEASON | DENSITY, type = "fixed", back.transform = TRUE) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

quinn.rstanarmNB |>
  fitted_draws(newdata = quinn) |>
  median_hdci() |>
  ggplot(aes(x = fSEASON, colour = DENSITY, y = .value)) +
  geom_pointrange(aes(ymin = .lower, ymax = .upper), position = position_dodge(width = 0.2)) +
  geom_line(position = position_dodge(width = 0.2)) +
  geom_point(data = quinn, aes(y = RECRUITS, x = fSEASON, colour = DENSITY), position = position_dodge(width = 0.2))
Warning: `fitted_draws` and `add_fitted_draws` are deprecated as their names were confusing.
- Use [add_]epred_draws() to get the expectation of the posterior predictive.
- Use [add_]linpred_draws() to get the distribution of the linear predictor.
- For example, you used [add_]fitted_draws(..., scale = "response"), which
  means you most likely want [add_]epred_draws(...).
NOTE: When updating to the new functions, note that the `model` parameter is now
  named `object` and the `n` parameter is now named `ndraws`.

quinn.brmsNB |>
  conditional_effects("fSEASON:DENSITY") |>
  plot(points = TRUE)

quinn.brmsNB |>
  ggpredict(~ fSEASON + DENSITY) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

quinn.brmsNB |>
  ggemmeans(~ fSEASON | DENSITY) |>
  plot(show_data = TRUE)
Data points may overlap. Use the `jitter` argument to add some amount of
  random variation to the location of data points and avoid overplotting.

quinn.brmsNB |>
  fitted_draws(newdata = quinn) |>
  median_hdci() |>
  ggplot(aes(x = fSEASON, colour = DENSITY, y = .value)) +
  geom_pointrange(aes(ymin = .lower, ymax = .upper), position = position_dodge(width = 0.2)) +
  geom_line(position = position_dodge(width = 0.2)) +
  geom_point(data = quinn, aes(y = RECRUITS, x = fSEASON, colour = DENSITY), position = position_dodge(width = 0.2))
Warning: `fitted_draws` and `add_fitted_draws` are deprecated as their names were confusing.
- Use [add_]epred_draws() to get the expectation of the posterior predictive.
- Use [add_]linpred_draws() to get the distribution of the linear predictor.
- For example, you used [add_]fitted_draws(..., scale = "response"), which
  means you most likely want [add_]epred_draws(...).
NOTE: When updating to the new functions, note that the `model` parameter is now
  named `object` and the `n` parameter is now named `ndraws`.

10 Model investigation

The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).

quinn.rstanarmNB |> summary()

Model Info:
 function:     stan_glm
 family:       neg_binomial_2 [log]
 formula:      RECRUITS ~ fSEASON * DENSITY
 algorithm:    sampling
 sample:       1800 (posterior sample size)
 priors:       see help('prior_summary')
 observations: 42
 predictors:   8

Estimates:
                           mean   sd   10%   50%   90%
(Intercept)               2.3    0.3  2.0   2.3   2.6 
fSEASONSummer             1.6    0.3  1.1   1.6   2.0 
fSEASONAutumn             0.7    0.3  0.2   0.7   1.1 
fSEASONWinter            -0.6    0.4 -1.1  -0.6  -0.1 
DENSITYLow                0.1    0.4 -0.4   0.1   0.6 
fSEASONSummer:DENSITYLow -0.9    0.5 -1.5  -0.9  -0.3 
fSEASONAutumn:DENSITYLow -0.2    0.5 -0.8  -0.2   0.5 
fSEASONWinter:DENSITYLow -0.9    0.7 -1.7  -0.9   0.0 
reciprocal_dispersion     4.0    1.3  2.6   3.8   5.7 

Fit Diagnostics:
           mean   sd   10%   50%   90%
mean_PPD 19.3    3.0 15.7  19.0  23.2 

The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).

MCMC diagnostics
                         mcse Rhat n_eff
(Intercept)              0.0  1.0  1811 
fSEASONSummer            0.0  1.0  1771 
fSEASONAutumn            0.0  1.0  1667 
fSEASONWinter            0.0  1.0  1872 
DENSITYLow               0.0  1.0  1817 
fSEASONSummer:DENSITYLow 0.0  1.0  1661 
fSEASONAutumn:DENSITYLow 0.0  1.0  1773 
fSEASONWinter:DENSITYLow 0.0  1.0  1940 
reciprocal_dispersion    0.0  1.0  1574 
mean_PPD                 0.1  1.0  1703 
log-posterior            0.1  1.0  1645 

For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).

Conclusions:

  • the intercept represents the estimated mean of the first combination of Season (Spring) and Density (High). On the link scale this is 2.32
  • the difference between Low and High adult density in spring is 1.56, although this is not significant
  • the difference between Spring and Summer for High adult density is 0.67
  • the difference between Spring and Autumn for High adult density is -0.58
  • the difference between Spring and Winter for High adult density is 0.12
  • if there were no interactions, we would expect the Low density Summer recruitment to be the additive of the main effects (Low and Summer). However, the modelled mean is 0.88 less than the additive effects would have expected. This value is significantly different to 0, indicating that there is evidence that the density effect in Summer is different to that in Spring.
  • the density effect in Autumn and Winter were not found to be significantly different from what you would expect from an additive model.
tidyMCMC(quinn.rstanarmNB$stanfit, estimate.method = "median", conf.int = TRUE, conf.method = "HPDinterval", rhat = TRUE, ess = TRUE)
# A tibble: 11 × 7
   term                     estimate std.error  conf.low conf.high  rhat   ess
   <chr>                       <dbl>     <dbl>     <dbl>     <dbl> <dbl> <int>
 1 (Intercept)                 2.32      0.251    1.84       2.83  1.00   1811
 2 fSEASONSummer               1.56      0.333    0.958      2.23  0.999  1771
 3 fSEASONAutumn               0.668     0.342    0.0161     1.35  1.00   1667
 4 fSEASONWinter              -0.582     0.375   -1.27       0.208 1.00   1872
 5 DENSITYLow                  0.116     0.368   -0.649      0.794 1.00   1817
 6 fSEASONSummer:DENSITYLow   -0.885     0.478   -1.76       0.100 1.00   1661
 7 fSEASONAutumn:DENSITYLow   -0.167     0.512   -1.15       0.914 1.00   1773
 8 fSEASONWinter:DENSITYLow   -0.888     0.658   -2.12       0.449 1.00   1940
 9 reciprocal_dispersion       4.00      1.25     1.81       6.63  1.00   1574
10 mean_PPD                   19.3       3.02    13.7       25.0   1.00   1703
11 log-posterior            -155.        2.35  -160.      -151.    1.00   1645

Conclusions:

See above

quinn.rstanarmNB$stanfit |>
  summarise_draws(
    median,
    HDInterval::hdi,
    rhat, length, ess_bulk, ess_tail
  )
# A tibble: 11 × 8
   variable              median    lower    upper  rhat length ess_bulk ess_tail
   <chr>                  <dbl>    <dbl>    <dbl> <dbl>  <dbl>    <dbl>    <dbl>
 1 (Intercept)            2.31   1.84e+0    2.83  1.00    1800    1824.    1637.
 2 fSEASONSummer          1.56   9.58e-1    2.23  0.999   1800    1787.    1652.
 3 fSEASONAutumn          0.670  1.61e-2    1.35  1.00    1800    1715.    1674.
 4 fSEASONWinter         -0.583 -1.27e+0    0.208 1.00    1800    1857.    1670.
 5 DENSITYLow             0.114 -6.49e-1    0.794 1.00    1800    1823.    1637.
 6 fSEASONSummer:DENS…   -0.900 -1.76e+0    0.100 1.00    1800    1667.    1594.
 7 fSEASONAutumn:DENS…   -0.150 -1.15e+0    0.914 1.00    1800    1783.    1786.
 8 fSEASONWinter:DENS…   -0.895 -2.12e+0    0.449 1.00    1800    1927.    1333.
 9 reciprocal_dispers…    3.79   1.81e+0    6.63  1.00    1800    1659.    1488.
10 mean_PPD              19.0    1.37e+1   25.0   1.00    1800    1736.    1598.
11 log-posterior       -155.    -1.60e+2 -151.    1.00    1800    1719.    1698.

We can also alter the CI level.

quinn.rstanarmNB$stanfit |>
  summarise_draws(
    median,
    ~ HDInterval::hdi(.x, credMass = 0.9),
    rhat, length, ess_bulk, ess_tail
  )
# A tibble: 11 × 8
   variable              median    lower    upper  rhat length ess_bulk ess_tail
   <chr>                  <dbl>    <dbl>    <dbl> <dbl>  <dbl>    <dbl>    <dbl>
 1 (Intercept)            2.31     1.91   2.73e+0 1.00    1800    1824.    1637.
 2 fSEASONSummer          1.56     1.06   2.15e+0 0.999   1800    1787.    1652.
 3 fSEASONAutumn          0.670    0.109  1.23e+0 1.00    1800    1715.    1674.
 4 fSEASONWinter         -0.583   -1.19   1.41e-2 1.00    1800    1857.    1670.
 5 DENSITYLow             0.114   -0.504  7.23e-1 1.00    1800    1823.    1637.
 6 fSEASONSummer:DENS…   -0.900   -1.67  -1.27e-1 1.00    1800    1667.    1594.
 7 fSEASONAutumn:DENS…   -0.150   -1.03   6.48e-1 1.00    1800    1783.    1786.
 8 fSEASONWinter:DENS…   -0.895   -1.86   2.97e-1 1.00    1800    1927.    1333.
 9 reciprocal_dispers…    3.79     2.16   6.06e+0 1.00    1800    1659.    1488.
10 mean_PPD              19.0     14.2    2.38e+1 1.00    1800    1736.    1598.
11 log-posterior       -155.    -158.    -1.52e+2 1.00    1800    1719.    1698.

Arguably, it would be better to back-transform to the ratio scale

quinn.rstanarmNB$stanfit |>
  summarise_draws(
    ~ median(exp(.x)),
    ~ HDInterval::hdi(exp(.x)),
    rhat, length, ess_bulk, ess_tail
  )
# A tibble: 11 × 8
   variable  `~median(exp(.x))`    lower    upper  rhat length ess_bulk ess_tail
   <chr>                  <dbl>    <dbl>    <dbl> <dbl>  <dbl>    <dbl>    <dbl>
 1 (Interce…           1.01e+ 1 5.70e+ 0 1.59e+ 1 1.00    1800    1824.    1637.
 2 fSEASONS…           4.75e+ 0 2.24e+ 0 8.49e+ 0 0.999   1800    1787.    1652.
 3 fSEASONA…           1.96e+ 0 8.65e- 1 3.56e+ 0 1.00    1800    1715.    1674.
 4 fSEASONW…           5.58e- 1 2.07e- 1 1.05e+ 0 1.00    1800    1857.    1670.
 5 DENSITYL…           1.12e+ 0 4.41e- 1 2.07e+ 0 1.00    1800    1823.    1637.
 6 fSEASONS…           4.06e- 1 1.08e- 1 9.18e- 1 1.00    1800    1667.    1594.
 7 fSEASONA…           8.61e- 1 2.38e- 1 2.06e+ 0 1.00    1800    1783.    1786.
 8 fSEASONW…           4.09e- 1 7.12e- 2 1.27e+ 0 1.00    1800    1927.    1333.
 9 reciproc…           4.41e+ 1 3.91e+ 0 6.19e+ 2 1.00    1800    1659.    1488.
10 mean_PPD            1.70e+ 8 6.43e+ 4 5.04e+10 1.00    1800    1736.    1598.
11 log-post…           6.51e-68 7.01e-74 9.01e-67 1.00    1800    1719.    1698.
quinn.rstanarmNB$stanfit |> as_draws_df()
# A draws_df: 600 iterations, 3 chains, and 11 variables
   (Intercept) fSEASONSummer fSEASONAutumn fSEASONWinter DENSITYLow
1          2.4           1.6          0.22         -0.69      0.138
2          2.2           1.3          0.96         -0.52      0.097
3          2.6           1.6          0.51         -0.89     -0.559
4          2.5           1.7          0.39         -0.83     -0.113
5          2.4           1.0          0.05         -0.65     -0.194
6          2.3           1.6          0.64         -0.79      0.122
7          2.4           1.4          0.60         -0.38      0.156
8          2.4           1.3          0.58         -0.57      0.191
9          2.3           1.4          0.81         -0.82      0.211
10         2.3           2.0          0.51         -0.27     -0.067
   fSEASONSummer:DENSITYLow fSEASONAutumn:DENSITYLow fSEASONWinter:DENSITYLow
1                     -1.18                    0.401                   -0.213
2                     -0.90                   -0.508                   -1.033
3                     -0.47                    0.063                   -0.107
4                     -1.11                   -0.104                   -0.616
5                     -0.22                    0.841                   -0.769
6                     -0.59                   -0.424                   -0.044
7                     -1.18                    0.213                   -1.514
8                     -0.78                   -0.844                   -0.799
9                     -0.41                   -0.962                   -1.039
10                    -1.33                    0.532                   -0.751
# ... with 1790 more draws, and 3 more variables
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}
quinn.rstanarmNB$stanfit |>
  as_draws_df() |>
  summarise_draws(
    median,
    ~ HDInterval::hdi(.x),
    rhat,
    ess_bulk
  )
# A tibble: 11 × 6
   variable                   median     lower    upper  rhat ess_bulk
   <chr>                       <dbl>     <dbl>    <dbl> <dbl>    <dbl>
 1 (Intercept)                 2.31     1.84      2.83  1.00     1824.
 2 fSEASONSummer               1.56     0.958     2.23  0.999    1787.
 3 fSEASONAutumn               0.670    0.0161    1.35  1.00     1715.
 4 fSEASONWinter              -0.583   -1.27      0.208 1.00     1857.
 5 DENSITYLow                  0.114   -0.649     0.794 1.00     1823.
 6 fSEASONSummer:DENSITYLow   -0.900   -1.76      0.100 1.00     1667.
 7 fSEASONAutumn:DENSITYLow   -0.150   -1.15      0.914 1.00     1783.
 8 fSEASONWinter:DENSITYLow   -0.895   -2.12      0.449 1.00     1927.
 9 reciprocal_dispersion       3.79     1.81      6.63  1.00     1659.
10 mean_PPD                   19.0     13.7      25.0   1.00     1736.
11 log-posterior            -155.    -160.     -151.    1.00     1719.
quinn.rstanarmNB$stanfit |>
  as_draws_df() |>
  exp() |>
  summarise_draws(
    median,
    ~ HDInterval::hdi(.x),
    rhat,
    ess_bulk
  )
# A tibble: 11 × 6
   variable                   median    lower    upper  rhat ess_bulk
   <chr>                       <dbl>    <dbl>    <dbl> <dbl>    <dbl>
 1 (Intercept)              1.01e+ 1 5.70e+ 0 1.59e+ 1 1.00     1824.
 2 fSEASONSummer            4.75e+ 0 2.24e+ 0 8.49e+ 0 0.999    1787.
 3 fSEASONAutumn            1.96e+ 0 8.65e- 1 3.56e+ 0 1.00     1715.
 4 fSEASONWinter            5.58e- 1 2.07e- 1 1.05e+ 0 1.00     1857.
 5 DENSITYLow               1.12e+ 0 4.41e- 1 2.07e+ 0 1.00     1823.
 6 fSEASONSummer:DENSITYLow 4.06e- 1 1.08e- 1 9.18e- 1 1.00     1667.
 7 fSEASONAutumn:DENSITYLow 8.61e- 1 2.38e- 1 2.06e+ 0 1.00     1783.
 8 fSEASONWinter:DENSITYLow 4.09e- 1 7.12e- 2 1.27e+ 0 1.00     1927.
 9 reciprocal_dispersion    4.41e+ 1 3.91e+ 0 6.19e+ 2 1.00     1659.
10 mean_PPD                 1.70e+ 8 6.43e+ 4 5.04e+10 1.00     1736.
11 log-posterior            6.51e-68 7.01e-74 9.01e-67 1.00     1719.

Due to the presence of a log transform in the predictor, it is better to use the regex version.

quinn.rstanarmNB |> get_variables()
 [1] "(Intercept)"              "fSEASONSummer"           
 [3] "fSEASONAutumn"            "fSEASONWinter"           
 [5] "DENSITYLow"               "fSEASONSummer:DENSITYLow"
 [7] "fSEASONAutumn:DENSITYLow" "fSEASONWinter:DENSITYLow"
 [9] "reciprocal_dispersion"    "accept_stat__"           
[11] "stepsize__"               "treedepth__"             
[13] "n_leapfrog__"             "divergent__"             
[15] "energy__"                
quinn.draw <- quinn.rstanarmNB |> gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE)
quinn.draw
# A tibble: 14,400 × 5
# Groups:   .variable [8]
   .chain .iteration .draw .variable   .value
    <int>      <int> <int> <chr>        <dbl>
 1      1          1     1 (Intercept)   2.38
 2      1          2     2 (Intercept)   2.23
 3      1          3     3 (Intercept)   2.55
 4      1          4     4 (Intercept)   2.46
 5      1          5     5 (Intercept)   2.39
 6      1          6     6 (Intercept)   2.28
 7      1          7     7 (Intercept)   2.36
 8      1          8     8 (Intercept)   2.40
 9      1          9     9 (Intercept)   2.29
10      1         10    10 (Intercept)   2.26
# ℹ 14,390 more rows
exceedP <- function(x, Val = 0) mean(x > Val)

quinn.rstanarmNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  mutate(.value = exp(.value)) |>
  summarise_draws(
    median,
    HDInterval::hdi,
    rhat,
    length,
    ess_bulk,
    ess_tail,
    ~ exceedP(.x, 1)
  )
# A tibble: 8 × 10
# Groups:   .variable [8]
  .variable         variable median  lower  upper  rhat length ess_bulk ess_tail
  <chr>             <chr>     <dbl>  <dbl>  <dbl> <dbl>  <dbl>    <dbl>    <dbl>
1 (Intercept)       .value   10.1   5.70   15.9   1.00    1800    1824.    1637.
2 DENSITYLow        .value    1.12  0.441   2.07  1.00    1800    1823.    1637.
3 fSEASONAutumn     .value    1.96  0.865   3.56  1.00    1800    1715.    1674.
4 fSEASONAutumn:DE… .value    0.861 0.238   2.06  1.00    1800    1783.    1786.
5 fSEASONSummer     .value    4.75  2.24    8.49  0.999   1800    1787.    1652.
6 fSEASONSummer:DE… .value    0.406 0.108   0.918 1.00    1800    1667.    1594.
7 fSEASONWinter     .value    0.558 0.207   1.05  1.00    1800    1857.    1670.
8 fSEASONWinter:DE… .value    0.409 0.0712  1.27  1.00    1800    1927.    1333.
# ℹ 1 more variable: `~exceedP(.x, 1)` <dbl>

We can then summarise this

quinn.draw |> median_hdci()
# A tibble: 8 × 7
  .variable                .value  .lower .upper .width .point .interval
  <chr>                     <dbl>   <dbl>  <dbl>  <dbl> <chr>  <chr>    
1 (Intercept)               2.31   1.81   2.80     0.95 median hdci     
2 DENSITYLow                0.114 -0.626  0.820    0.95 median hdci     
3 fSEASONAutumn             0.670  0.0161 1.35     0.95 median hdci     
4 fSEASONAutumn:DENSITYLow -0.150 -1.29   0.785    0.95 median hdci     
5 fSEASONSummer             1.56   0.965  2.24     0.95 median hdci     
6 fSEASONSummer:DENSITYLow -0.900 -1.80   0.0689   0.95 median hdci     
7 fSEASONWinter            -0.583 -1.27   0.208    0.95 median hdci     
8 fSEASONWinter:DENSITYLow -0.895 -2.16   0.407    0.95 median hdci     

We could alternatively express the parameters on the response scale.

quinn.draw |> median_hdci(exp(.value))
# A tibble: 8 × 7
  .variable                `exp(.value)` .lower .upper .width .point .interval
  <chr>                            <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
1 (Intercept)                     10.1   5.70   15.9     0.95 median hdci     
2 DENSITYLow                       1.12  0.466   2.10    0.95 median hdci     
3 fSEASONAutumn                    1.96  0.788   3.49    0.95 median hdci     
4 fSEASONAutumn:DENSITYLow         0.861 0.238   2.06    0.95 median hdci     
5 fSEASONSummer                    4.75  2.24    8.49    0.95 median hdci     
6 fSEASONSummer:DENSITYLow         0.406 0.108   0.918   0.95 median hdci     
7 fSEASONWinter                    0.558 0.207   1.05    0.95 median hdci     
8 fSEASONWinter:DENSITYLow         0.409 0.0835  1.28    0.95 median hdci     
quinn.rstanarmNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  stat_halfeye(aes(x = .value, y = .variable)) +
  facet_wrap(~.variable, scales = "free")

quinn.rstanarmNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  geom_vline(xintercept = 0, linetype = "dashed") +
  stat_halfeye(aes(x = .value, y = .variable)) +
  theme_classic()

We could alternatively express the parameters on the response scale.

quinn.rstanarmNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  group_by(.variable) |>
  mutate(.value = exp(.value)) |>
  median_hdci()
# A tibble: 8 × 7
  .variable                .value .lower .upper .width .point .interval
  <chr>                     <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
1 (Intercept)              10.1   5.70   15.9     0.95 median hdci     
2 DENSITYLow                1.12  0.466   2.10    0.95 median hdci     
3 fSEASONAutumn             1.96  0.788   3.49    0.95 median hdci     
4 fSEASONAutumn:DENSITYLow  0.861 0.238   2.06    0.95 median hdci     
5 fSEASONSummer             4.75  2.24    8.49    0.95 median hdci     
6 fSEASONSummer:DENSITYLow  0.406 0.108   0.918   0.95 median hdci     
7 fSEASONWinter             0.558 0.207   1.05    0.95 median hdci     
8 fSEASONWinter:DENSITYLow  0.409 0.0835  1.28    0.95 median hdci     
quinn.rstanarmNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  mutate(.value = exp(.value)) |>
  ggplot() +
  geom_vline(xintercept = 1, linetype = "dashed") +
  stat_halfeye(aes(x = .value, y = .variable)) +
  scale_x_continuous("", trans = scales::log2_trans(), breaks = unique(as.vector(2^(0:4 %o% c(-1, 1))))) +
  theme_classic()

quinn.rstanarmNB |> plot(plotfun = "mcmc_intervals")

## Link scale
quinn.rstanarmNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  stat_slab(aes(
    x = .value, y = .variable,
    fill = stat(ggdist::cut_cdf_qi(cdf,
      .width = c(0.5, 0.8, 0.95),
      labels = scales::percent_format()
    ))
  ), color = "black") +
  geom_vline(xintercept = 0, linetype = "dashed") +
  scale_fill_brewer("Interval", direction = -1, na.translate = FALSE)
Warning: `stat(ggdist::cut_cdf_qi(cdf, .width = c(0.5, 0.8, 0.95), labels =
scales::percent_format()))` was deprecated in ggplot2 3.4.0.
ℹ Please use `after_stat(ggdist::cut_cdf_qi(cdf, .width = c(0.5, 0.8, 0.95),
  labels = scales::percent_format()))` instead.

## Fractional scale
quinn.rstanarmNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  mutate(.value = exp(.value)) |>
  ggplot() +
  stat_slab(aes(
    x = .value, y = .variable,
    fill = stat(ggdist::cut_cdf_qi(cdf,
      .width = c(0.5, 0.8, 0.95),
      labels = scales::percent_format()
    ))
  ), color = "black") +
  geom_vline(xintercept = 1, linetype = "dashed") +
  scale_fill_brewer("Interval", direction = -1, na.translate = FALSE) +
  scale_x_continuous(trans = scales::log2_trans())

quinn.rstanarmNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  stat_halfeye(aes(x = .value, y = .variable)) +
  facet_wrap(~.variable, scales = "free")

quinn.rstanarmNB |>
  gather_draws(`.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  stat_halfeye(aes(x = .value, y = .variable)) +
  geom_vline(xintercept = 0, linetype = "dashed")

quinn.rstanarmNB |>
  gather_draws(`.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  stat_halfeye(aes(x = exp(.value), y = .variable)) +
  geom_vline(xintercept = 1, linetype = "dashed") +
  scale_x_continuous(trans = scales::log2_trans())

quinn.rstanarmNB |>
  gather_draws(`.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  geom_density_ridges(aes(x = .value, y = .variable), alpha = 0.4) +
  geom_vline(xintercept = 0, linetype = "dashed")
Picking joint bandwidth of 0.0865

## Or on a fractional scale
quinn.rstanarmNB |>
  gather_draws(`.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  geom_density_ridges_gradient(
    aes(
      x = exp(.value),
      y = .variable,
      fill = stat(x)
    ),
    alpha = 0.4, colour = "white",
    quantile_lines = TRUE,
    quantiles = c(0.025, 0.975)
  ) +
  geom_vline(xintercept = 1, linetype = "dashed") +
  scale_x_continuous(trans = scales::log2_trans()) +
  scale_fill_viridis_c(option = "C")
Picking joint bandwidth of 0.125

This is purely a graphical depiction on the posteriors.

quinn.rstanarmNB |> tidy_draws()
# A tibble: 1,800 × 18
   .chain .iteration .draw `(Intercept)` fSEASONSummer fSEASONAutumn
    <int>      <int> <int>         <dbl>         <dbl>         <dbl>
 1      1          1     1          2.38          1.60        0.221 
 2      1          2     2          2.23          1.28        0.956 
 3      1          3     3          2.55          1.60        0.514 
 4      1          4     4          2.46          1.72        0.389 
 5      1          5     5          2.39          1.02        0.0498
 6      1          6     6          2.28          1.61        0.643 
 7      1          7     7          2.36          1.36        0.596 
 8      1          8     8          2.40          1.33        0.580 
 9      1          9     9          2.29          1.43        0.810 
10      1         10    10          2.26          1.99        0.509 
# ℹ 1,790 more rows
# ℹ 12 more variables: fSEASONWinter <dbl>, DENSITYLow <dbl>,
#   `fSEASONSummer:DENSITYLow` <dbl>, `fSEASONAutumn:DENSITYLow` <dbl>,
#   `fSEASONWinter:DENSITYLow` <dbl>, reciprocal_dispersion <dbl>,
#   accept_stat__ <dbl>, stepsize__ <dbl>, treedepth__ <dbl>,
#   n_leapfrog__ <dbl>, divergent__ <dbl>, energy__ <dbl>
quinn.rstanarmNB |> spread_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE)
# A tibble: 1,800 × 11
   .chain .iteration .draw `(Intercept)` fSEASONSummer fSEASONAutumn
    <int>      <int> <int>         <dbl>         <dbl>         <dbl>
 1      1          1     1          2.38          1.60        0.221 
 2      1          2     2          2.23          1.28        0.956 
 3      1          3     3          2.55          1.60        0.514 
 4      1          4     4          2.46          1.72        0.389 
 5      1          5     5          2.39          1.02        0.0498
 6      1          6     6          2.28          1.61        0.643 
 7      1          7     7          2.36          1.36        0.596 
 8      1          8     8          2.40          1.33        0.580 
 9      1          9     9          2.29          1.43        0.810 
10      1         10    10          2.26          1.99        0.509 
# ℹ 1,790 more rows
# ℹ 5 more variables: fSEASONWinter <dbl>, DENSITYLow <dbl>,
#   `fSEASONSummer:DENSITYLow` <dbl>, `fSEASONAutumn:DENSITYLow` <dbl>,
#   `fSEASONWinter:DENSITYLow` <dbl>
quinn.rstanarmNB |>
  posterior_samples() |>
  as_tibble()
Warning: Method 'posterior_samples' is deprecated. Please see ?as_draws for
recommended alternatives.
# A tibble: 1,800 × 9
   `(Intercept)` fSEASONSummer fSEASONAutumn fSEASONWinter DENSITYLow
           <dbl>         <dbl>         <dbl>         <dbl>      <dbl>
 1          2.38          1.60        0.221         -0.695     0.138 
 2          2.23          1.28        0.956         -0.523     0.0968
 3          2.55          1.60        0.514         -0.890    -0.559 
 4          2.46          1.72        0.389         -0.827    -0.113 
 5          2.39          1.02        0.0498        -0.654    -0.194 
 6          2.28          1.61        0.643         -0.792     0.122 
 7          2.36          1.36        0.596         -0.381     0.156 
 8          2.40          1.33        0.580         -0.566     0.191 
 9          2.29          1.43        0.810         -0.823     0.211 
10          2.26          1.99        0.509         -0.269    -0.0674
# ℹ 1,790 more rows
# ℹ 4 more variables: `fSEASONSummer:DENSITYLow` <dbl>,
#   `fSEASONAutumn:DENSITYLow` <dbl>, `fSEASONWinter:DENSITYLow` <dbl>,
#   reciprocal_dispersion <dbl>

Unfortunately, \(R^2\) calculations for models other than Gaussian and Binomial have not yet been implemented for rstanarm models yet.

# quinn.rstanarmNB |> bayes_R2() |> median_hdci

The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).

quinn.brmsNB |> summary()
 Family: negbinomial 
  Links: mu = log; shape = identity 
Formula: RECRUITS ~ fSEASON * DENSITY 
   Data: quinn (Number of observations: 42) 
  Draws: 3 chains, each with iter = 5000; warmup = 2500; thin = 5;
         total post-warmup draws = 1500

Regression Coefficients:
                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
Intercept                    2.41      0.20     2.04     2.83 1.00     1116
fSEASONSummer                1.41      0.25     0.90     1.87 1.00     1386
fSEASONAutumn                0.56      0.26     0.04     1.05 1.00     1311
fSEASONWinter               -0.68      0.29    -1.25    -0.11 1.00     1507
DENSITYLow                  -0.06      0.26    -0.58     0.45 1.00     1220
fSEASONSummer:DENSITYLow    -0.64      0.34    -1.27     0.03 1.00     1393
fSEASONAutumn:DENSITYLow    -0.01      0.38    -0.74     0.78 1.00     1277
fSEASONWinter:DENSITYLow    -0.62      0.48    -1.57     0.34 1.00     1548
                         Tail_ESS
Intercept                    1097
fSEASONSummer                1501
fSEASONAutumn                1414
fSEASONWinter                1519
DENSITYLow                   1340
fSEASONSummer:DENSITYLow     1347
fSEASONAutumn:DENSITYLow     1382
fSEASONWinter:DENSITYLow     1458

Further Distributional Parameters:
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
shape     6.79      3.09     2.84    14.45 1.00     1468     1482

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

Conclusions:

  • the intercept represents the estimated mean of the first combination of Season (Spring) and Density (High). On the link scale this is 2.41
  • the difference between Low and High adult density in spring is 1.41, although this is not significant
  • the difference between Spring and Summer for High adult density is 0.56
  • the difference between Spring and Autumn for High adult density is -0.68
  • the difference between Spring and Winter for High adult density is -0.06
  • if there were no interactions, we would expect the Low density Summer recruitment to be the additive of the main effects (Low and Summer). However, the modelled mean is 0.64 less than the additive effects would have expected. This value is significantly different to 0, indicating that there is evidence that the density effect in Summer is different to that in Spring.
  • the density effect in Autumn and Winter were not found to be significantly different from what you would expect from an additive model.
quinn.brmsNB$fit |>
  tidyMCMC(
    estimate.method = "median",
    conf.int = TRUE,
    conf.method = "HPDinterval",
    rhat = TRUE,
    ess = TRUE
  )
# A tibble: 11 × 7
   term                        estimate std.error conf.low conf.high  rhat   ess
   <chr>                          <dbl>     <dbl>    <dbl>     <dbl> <dbl> <int>
 1 b_Intercept                  2.41       0.196    2.01      2.79   1.00   1108
 2 b_fSEASONSummer              1.41       0.255    0.900     1.87   0.999  1368
 3 b_fSEASONAutumn              0.559      0.261    0.0407    1.05   1.00   1317
 4 b_fSEASONWinter             -0.677      0.287   -1.28     -0.149  1.00   1517
 5 b_DENSITYLow                -0.0554     0.257   -0.532     0.483  1.00   1206
 6 b_fSEASONSummer:DENSITYLow  -0.644      0.342   -1.27      0.0317 0.999  1400
 7 b_fSEASONAutumn:DENSITYLow  -0.00695    0.375   -0.689     0.802  1.00   1268
 8 b_fSEASONWinter:DENSITYLow  -0.623      0.484   -1.63      0.257  0.999  1542
 9 shape                        6.79       3.09     2.37     12.7    1.00   1433
10 Intercept                    2.64       0.0816   2.47      2.79   1.000  1555
11 lprior                     -16.5        0.825  -18.1     -14.9    1.00   1393

Conclusions:

see above

quinn.brmsNB |> as_draws_df()
# A draws_df: 500 iterations, 3 chains, and 12 variables
   b_Intercept b_fSEASONSummer b_fSEASONAutumn b_fSEASONWinter b_DENSITYLow
1          2.7            1.16           0.520           -0.71       -0.170
2          2.8            1.04           0.065           -0.94       -0.314
3          2.8            1.09           0.305           -0.97       -1.017
4          2.9            1.11           0.379           -0.54       -0.797
5          2.5            1.25           0.567           -1.28       -0.299
6          2.8            1.09          -0.046           -0.84       -0.224
7          2.5            1.38           0.135           -0.65        0.022
8          2.8            0.96           0.230           -1.24       -0.304
9          2.3            1.51           0.895           -0.51        0.093
10         2.6            1.43           0.645           -1.02       -0.177
   b_fSEASONSummer:DENSITYLow b_fSEASONAutumn:DENSITYLow
1                       -0.47                      0.464
2                       -0.29                      0.311
3                        0.47                      0.819
4                        0.18                      0.225
5                       -0.46                     -0.037
6                       -0.79                      0.342
7                       -0.40                      0.177
8                       -0.29                      0.191
9                       -0.86                     -0.234
10                      -1.19                     -0.366
   b_fSEASONWinter:DENSITYLow
1                       -0.44
2                       -0.39
3                        0.54
4                       -0.93
5                       -0.23
6                       -0.21
7                       -0.53
8                       -0.90
9                       -1.96
10                      -0.18
# ... with 1490 more draws, and 4 more variables
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}
quinn.brmsNB |>
  as_draws_df() |>
  summarise_draws(
    median,
    ~ HDInterval::hdi(.x),
    rhat,
    ess_bulk, ess_tail
  )
# A tibble: 12 × 7
   variable                     median    lower    upper  rhat ess_bulk ess_tail
   <chr>                         <dbl>    <dbl>    <dbl> <dbl>    <dbl>    <dbl>
 1 b_Intercept                 2.41e+0  2.01e+0  2.79e+0 1.00     1117.    1097.
 2 b_fSEASONSummer             1.42e+0  9.00e-1  1.87e+0 0.999    1386.    1501.
 3 b_fSEASONAutumn             5.62e-1  4.07e-2  1.05e+0 1.00     1310.    1414.
 4 b_fSEASONWinter            -6.85e-1 -1.28e+0 -1.49e-1 1.00     1507.    1519.
 5 b_DENSITYLow               -4.84e-2 -5.32e-1  4.83e-1 1.00     1219.    1340.
 6 b_fSEASONSummer:DENSITYLow -6.49e-1 -1.27e+0  3.17e-2 0.999    1393.    1347.
 7 b_fSEASONAutumn:DENSITYLow -1.14e-2 -6.89e-1  8.02e-1 1.00     1277.    1382.
 8 b_fSEASONWinter:DENSITYLow -6.28e-1 -1.63e+0  2.57e-1 1.00     1548.    1458.
 9 shape                       6.15e+0  2.37e+0  1.27e+1 1.00     1468.    1482.
10 Intercept                   2.64e+0  2.47e+0  2.79e+0 1.000    1567.    1453.
11 lprior                     -1.64e+1 -1.81e+1 -1.49e+1 1.00     1406.    1428.
12 lp__                       -1.57e+2 -1.62e+2 -1.53e+2 1.00     1442.    1419.
quinn.brmsNB |>
  as_draws_df() |>
  exp() |>
  summarise_draws(
    median,
    HDInterval::hdi,
    rhat,
    length,
    ess_bulk, ess_tail
  )
# A tibble: 12 × 8
   variable              median    lower    upper  rhat length ess_bulk ess_tail
   <chr>                  <dbl>    <dbl>    <dbl> <dbl>  <dbl>    <dbl>    <dbl>
 1 b_Intercept         1.11e+ 1 7.32e+ 0 1.61e+ 1 1.00    1500    1117.    1097.
 2 b_fSEASONSummer     4.12e+ 0 2.34e+ 0 6.30e+ 0 0.999   1500    1386.    1501.
 3 b_fSEASONAutumn     1.75e+ 0 9.63e- 1 2.73e+ 0 1.00    1500    1310.    1414.
 4 b_fSEASONWinter     5.04e- 1 2.55e- 1 8.22e- 1 1.00    1500    1507.    1519.
 5 b_DENSITYLow        9.53e- 1 5.05e- 1 1.48e+ 0 1.00    1500    1219.    1340.
 6 b_fSEASONSummer:DE… 5.23e- 1 2.63e- 1 9.93e- 1 0.999   1500    1393.    1347.
 7 b_fSEASONAutumn:DE… 9.89e- 1 4.28e- 1 1.95e+ 0 1.00    1500    1277.    1382.
 8 b_fSEASONWinter:DE… 5.34e- 1 1.54e- 1 1.19e+ 0 1.00    1500    1548.    1458.
 9 shape               4.69e+ 2 5.25e+ 0 2.46e+ 5 1.00    1500    1468.    1482.
10 Intercept           1.41e+ 1 1.18e+ 1 1.63e+ 1 1.000   1500    1567.    1453.
11 lprior              7.65e- 8 3.68e- 9 2.29e- 7 1.00    1500    1406.    1428.
12 lp__                9.12e-69 3.97e-73 1.47e-67 1.00    1500    1442.    1419.

Due to the presence of a log transform in the predictor, it is better to use the regex version.

quinn.brmsNB |> get_variables()
 [1] "b_Intercept"                "b_fSEASONSummer"           
 [3] "b_fSEASONAutumn"            "b_fSEASONWinter"           
 [5] "b_DENSITYLow"               "b_fSEASONSummer:DENSITYLow"
 [7] "b_fSEASONAutumn:DENSITYLow" "b_fSEASONWinter:DENSITYLow"
 [9] "shape"                      "Intercept"                 
[11] "lprior"                     "lp__"                      
[13] "accept_stat__"              "stepsize__"                
[15] "treedepth__"                "n_leapfrog__"              
[17] "divergent__"                "energy__"                  
quinn.draw <- quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE)
quinn.draw
# A tibble: 10,500 × 5
# Groups:   .variable [7]
   .chain .iteration .draw .variable       .value
    <int>      <int> <int> <chr>            <dbl>
 1      1          1     1 b_fSEASONSummer  1.16 
 2      1          2     2 b_fSEASONSummer  1.04 
 3      1          3     3 b_fSEASONSummer  1.09 
 4      1          4     4 b_fSEASONSummer  1.11 
 5      1          5     5 b_fSEASONSummer  1.25 
 6      1          6     6 b_fSEASONSummer  1.09 
 7      1          7     7 b_fSEASONSummer  1.38 
 8      1          8     8 b_fSEASONSummer  0.957
 9      1          9     9 b_fSEASONSummer  1.51 
10      1         10    10 b_fSEASONSummer  1.43 
# ℹ 10,490 more rows
quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  mutate(.value = exp(.value)) |>
  summarise_draws(
    median,
    ~ HDInterval::hdi(.x, credMass = 0.95),
    rhat,
    length,
    ess_bulk, ess_tail
  )
# A tibble: 7 × 9
# Groups:   .variable [7]
  .variable           variable median lower upper  rhat length ess_bulk ess_tail
  <chr>               <chr>     <dbl> <dbl> <dbl> <dbl>  <dbl>    <dbl>    <dbl>
1 b_DENSITYLow        .value    0.953 0.505 1.48  1.00    1500    1219.    1340.
2 b_fSEASONAutumn     .value    1.75  0.963 2.73  1.00    1500    1310.    1414.
3 b_fSEASONAutumn:DE… .value    0.989 0.428 1.95  1.00    1500    1277.    1382.
4 b_fSEASONSummer     .value    4.12  2.34  6.30  0.999   1500    1386.    1501.
5 b_fSEASONSummer:DE… .value    0.523 0.263 0.993 0.999   1500    1393.    1347.
6 b_fSEASONWinter     .value    0.504 0.255 0.822 1.00    1500    1507.    1519.
7 b_fSEASONWinter:DE… .value    0.534 0.154 1.19  1.00    1500    1548.    1458.
exceedP <- function(x, Val = 0) mean(x > Val)
quinn.brmsNB |>
  tidy_draws() |>
  exp() |>
  dplyr::select(starts_with("b_")) |>
  summarise_draws(
    median,
    ~ HDInterval::hdi(.x, credMass = 0.9),
    rhat,
    ess_bulk, ess_tail,
    ~ exceedP(.x, 1)
  )
# A tibble: 8 × 8
  variable         median lower  upper  rhat ess_bulk ess_tail `~exceedP(.x, 1)`
  <chr>             <dbl> <dbl>  <dbl> <dbl>    <dbl>    <dbl>             <dbl>
1 b_Intercept      11.1   7.46  14.7   1.000    1100.     975.            1     
2 b_fSEASONSummer   4.12  2.46   5.86  1.000    1346.    1460.            1     
3 b_fSEASONAutumn   1.75  1.10   2.58  1.000    1303.    1408.            0.983 
4 b_fSEASONWinter   0.504 0.292  0.769 0.999    1504.    1512.            0.0107
5 b_DENSITYLow      0.953 0.569  1.38  1.000    1210.    1332.            0.417 
6 b_fSEASONSummer…  0.523 0.271  0.859 0.999    1391.    1253.            0.0347
7 b_fSEASONAutumn…  0.989 0.471  1.66  1.00     1274.    1378.            0.485 
8 b_fSEASONWinter…  0.534 0.176  1.01  0.999    1537.    1453.            0.0947

We can then summarise this

quinn.draw |> median_hdci()
# A tibble: 7 × 7
  .variable                   .value  .lower   .upper .width .point .interval
  <chr>                        <dbl>   <dbl>    <dbl>  <dbl> <chr>  <chr>    
1 b_DENSITYLow               -0.0484 -0.532   0.483     0.95 median hdci     
2 b_fSEASONAutumn             0.562   0.0407  1.05      0.95 median hdci     
3 b_fSEASONAutumn:DENSITYLow -0.0114 -0.693   0.798     0.95 median hdci     
4 b_fSEASONSummer             1.42    0.898   1.87      0.95 median hdci     
5 b_fSEASONSummer:DENSITYLow -0.649  -1.30    0.00510   0.95 median hdci     
6 b_fSEASONWinter            -0.685  -1.30   -0.163     0.95 median hdci     
7 b_fSEASONWinter:DENSITYLow -0.628  -1.63    0.257     0.95 median hdci     

We could alternatively express the parameters on the response scale.

quinn.draw |>
  median_hdci(exp(.value))
# A tibble: 7 × 7
  .variable                  `exp(.value)` .lower .upper .width .point .interval
  <chr>                              <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
1 b_DENSITYLow                       0.953  0.505  1.48    0.95 median hdci     
2 b_fSEASONAutumn                    1.75   0.992  2.78    0.95 median hdci     
3 b_fSEASONAutumn:DENSITYLow         0.989  0.428  1.95    0.95 median hdci     
4 b_fSEASONSummer                    4.12   2.34   6.30    0.95 median hdci     
5 b_fSEASONSummer:DENSITYLow         0.523  0.273  1.01    0.95 median hdci     
6 b_fSEASONWinter                    0.504  0.272  0.847   0.95 median hdci     
7 b_fSEASONWinter:DENSITYLow         0.534  0.154  1.19    0.95 median hdci     
quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  geom_vline(xintercept = 0, linetype = "dashed") +
  stat_slab(aes(
    x = .value, y = .variable,
    fill = stat(ggdist::cut_cdf_qi(cdf,
      .width = c(0.5, 0.8, 0.95),
      labels = scales::percent_format()
    ))
  ), color = "black") +
  scale_fill_brewer("Interval", direction = -1, na.translate = FALSE)

quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  geom_vline(xintercept = 0, linetype = "dashed") +
  stat_halfeye(aes(x = .value, y = .variable)) +
  theme_classic()

We could alternatively express the parameters on the response scale.

quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  group_by(.variable) |>
  mutate(.value = exp(.value)) |>
  median_hdci()
# A tibble: 7 × 7
  .variable                  .value .lower .upper .width .point .interval
  <chr>                       <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
1 b_DENSITYLow                0.953  0.505  1.48    0.95 median hdci     
2 b_fSEASONAutumn             1.75   0.992  2.78    0.95 median hdci     
3 b_fSEASONAutumn:DENSITYLow  0.989  0.428  1.95    0.95 median hdci     
4 b_fSEASONSummer             4.12   2.34   6.30    0.95 median hdci     
5 b_fSEASONSummer:DENSITYLow  0.523  0.273  1.01    0.95 median hdci     
6 b_fSEASONWinter             0.504  0.272  0.847   0.95 median hdci     
7 b_fSEASONWinter:DENSITYLow  0.534  0.154  1.19    0.95 median hdci     
quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  mutate(.value = exp(.value)) |>
  ggplot() +
  geom_vline(xintercept = 1, linetype = "dashed") +
  stat_slab(aes(
    x = .value, y = .variable,
    fill = stat(ggdist::cut_cdf_qi(cdf,
      .width = c(0.5, 0.8, 0.95),
      labels = scales::percent_format()
    ))
  ), color = "black") +
  scale_x_continuous("", trans = scales::log2_trans(), breaks = unique(as.vector(2^(0:4 %o% c(-1, 1))))) +
  scale_fill_brewer("Interval", direction = -1, na.translate = FALSE) +
  theme_classic()

Conclusions:

  • the estimated mean (expected number of newly recruited barnacles) on the ALG1 surface is -0.05. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 0.95.
  • the estimated effect of ALG2 vs ALG1 is 0.56 (median) with a standard error of 0.04. The 95% credibility intervals indicate that we are 95% confident that the effect is between 1.05 and 0.95 - e.g. there is a significant positive effect. When back-transformed onto the response scale, we see that barnacle recruitment on ALG2 is 1.75 times higher than that on ALG1. This represents a 75% increase in barnacle recruitment.
  • the estimated effect of NB and S are -0.01 and 1.42 respectively, which equate to 1.01 and 0.24 fold declines respectively.
  • Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
quinn.brmsNB$fit |> plot(type = "intervals")
'pars' not specified. Showing first 10 parameters by default.
ci_level: 0.8 (80% intervals)
outer_level: 0.95 (95% intervals)

## Link scale
quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  stat_slab(aes(
    x = .value, y = .variable,
    fill = stat(ggdist::cut_cdf_qi(cdf,
      .width = c(0.5, 0.8, 0.95),
      labels = scales::percent_format()
    ))
  ), color = "black") +
  geom_vline(xintercept = 0, linetype = "dashed") +
  scale_fill_brewer("Interval", direction = -1, na.translate = FALSE)

## Fractional scale
quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  mutate(.value = exp(.value)) |>
  ggplot() +
  stat_slab(aes(
    x = .value, y = .variable,
    fill = stat(ggdist::cut_cdf_qi(cdf,
      .width = c(0.5, 0.8, 0.95),
      labels = scales::percent_format()
    ))
  ), color = "black") +
  geom_vline(xintercept = 1, linetype = "dashed") +
  scale_fill_brewer("Interval", direction = -1, na.translate = FALSE) +
  scale_x_continuous(trans = scales::log2_trans())

quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  stat_halfeye(aes(x = .value, y = .variable)) +
  facet_wrap(~.variable, scales = "free")

quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  stat_halfeye(aes(x = .value, y = .variable)) +
  geom_vline(xintercept = 0, linetype = "dashed")

quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  stat_halfeye(aes(x = exp(.value), y = .variable)) +
  geom_vline(xintercept = 1, linetype = "dashed") +
  scale_x_continuous(trans = scales::log2_trans())

quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  geom_density_ridges(aes(x = .value, y = .variable), alpha = 0.4) +
  geom_vline(xintercept = 0, linetype = "dashed")
Picking joint bandwidth of 0.0663

## Or on a fractional scale
quinn.brmsNB |>
  gather_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE) |>
  ggplot() +
  geom_density_ridges_gradient(
    aes(
      x = exp(.value),
      y = .variable,
      fill = stat(x)
    ),
    alpha = 0.4, colour = "white",
    quantile_lines = TRUE,
    quantiles = c(0.025, 0.975)
  ) +
  geom_vline(xintercept = 1, linetype = "dashed") +
  scale_x_continuous(trans = scales::log2_trans()) +
  scale_fill_viridis_c(option = "C")
Picking joint bandwidth of 0.0956

This is purely a graphical depiction on the posteriors.

quinn.brmsNB |> tidy_draws()
# A tibble: 1,500 × 21
   .chain .iteration .draw b_Intercept b_fSEASONSummer b_fSEASONAutumn
    <int>      <int> <int>       <dbl>           <dbl>           <dbl>
 1      1          1     1        2.72           1.16           0.520 
 2      1          2     2        2.81           1.04           0.0650
 3      1          3     3        2.79           1.09           0.305 
 4      1          4     4        2.89           1.11           0.379 
 5      1          5     5        2.47           1.25           0.567 
 6      1          6     6        2.81           1.09          -0.0464
 7      1          7     7        2.55           1.38           0.135 
 8      1          8     8        2.83           0.957          0.230 
 9      1          9     9        2.32           1.51           0.895 
10      1         10    10        2.55           1.43           0.645 
# ℹ 1,490 more rows
# ℹ 15 more variables: b_fSEASONWinter <dbl>, b_DENSITYLow <dbl>,
#   `b_fSEASONSummer:DENSITYLow` <dbl>, `b_fSEASONAutumn:DENSITYLow` <dbl>,
#   `b_fSEASONWinter:DENSITYLow` <dbl>, shape <dbl>, Intercept <dbl>,
#   lprior <dbl>, lp__ <dbl>, accept_stat__ <dbl>, stepsize__ <dbl>,
#   treedepth__ <dbl>, n_leapfrog__ <dbl>, divergent__ <dbl>, energy__ <dbl>
quinn.brmsNB |> spread_draws(`.Intercept.*|.*fSEASON.*|.*DENSITY.*`, regex = TRUE)
# A tibble: 1,500 × 10
   .chain .iteration .draw b_fSEASONSummer b_fSEASONAutumn b_fSEASONWinter
    <int>      <int> <int>           <dbl>           <dbl>           <dbl>
 1      1          1     1           1.16           0.520           -0.713
 2      1          2     2           1.04           0.0650          -0.945
 3      1          3     3           1.09           0.305           -0.974
 4      1          4     4           1.11           0.379           -0.541
 5      1          5     5           1.25           0.567           -1.28 
 6      1          6     6           1.09          -0.0464          -0.837
 7      1          7     7           1.38           0.135           -0.646
 8      1          8     8           0.957          0.230           -1.24 
 9      1          9     9           1.51           0.895           -0.512
10      1         10    10           1.43           0.645           -1.02 
# ℹ 1,490 more rows
# ℹ 4 more variables: b_DENSITYLow <dbl>, `b_fSEASONSummer:DENSITYLow` <dbl>,
#   `b_fSEASONAutumn:DENSITYLow` <dbl>, `b_fSEASONWinter:DENSITYLow` <dbl>
quinn.brmsNB |>
  posterior_samples() |>
  as_tibble()
Warning: Method 'posterior_samples' is deprecated. Please see ?as_draws for
recommended alternatives.
# A tibble: 1,500 × 12
   b_Intercept b_fSEASONSummer b_fSEASONAutumn b_fSEASONWinter b_DENSITYLow
         <dbl>           <dbl>           <dbl>           <dbl>        <dbl>
 1        2.72           1.16           0.520           -0.713      -0.170 
 2        2.81           1.04           0.0650          -0.945      -0.314 
 3        2.79           1.09           0.305           -0.974      -1.02  
 4        2.89           1.11           0.379           -0.541      -0.797 
 5        2.47           1.25           0.567           -1.28       -0.299 
 6        2.81           1.09          -0.0464          -0.837      -0.224 
 7        2.55           1.38           0.135           -0.646       0.0223
 8        2.83           0.957          0.230           -1.24       -0.304 
 9        2.32           1.51           0.895           -0.512       0.0934
10        2.55           1.43           0.645           -1.02       -0.177 
# ℹ 1,490 more rows
# ℹ 7 more variables: `b_fSEASONSummer:DENSITYLow` <dbl>,
#   `b_fSEASONAutumn:DENSITYLow` <dbl>, `b_fSEASONWinter:DENSITYLow` <dbl>,
#   shape <dbl>, Intercept <dbl>, lprior <dbl>, lp__ <dbl>
quinn.brmsNB |>
  bayes_R2(summary = FALSE) |>
  median_hdci()
          y      ymin      ymax .width .point .interval
1 0.7272419 0.5231952 0.8084454   0.95 median      hdci

Region of Practical Equivalence

0.1 * log(sd(quinn$RECRUITS))
[1] 0.2754809
quinn.brmsNB |> rope(range = c(-0.28, 0.28))
# Proportion of samples inside the ROPE [-0.28, 0.28]:

Parameter                | inside ROPE
--------------------------------------
Intercept                |      0.00 %
fSEASONSummer            |      0.00 %
fSEASONAutumn            |     12.64 %
fSEASONWinter            |      5.69 %
DENSITYLow               |     74.30 %
fSEASONSummer:DENSITYLow |     12.01 %
fSEASONAutumn:DENSITYLow |     57.65 %
fSEASONWinter:DENSITYLow |     20.37 %
rope(quinn.brmsNB, range = c(-0.28, 0.28)) |> plot()

## Or based on fractional scale
quinn.brmsNB |>
  as_draws_df("^b_fSEASON.*|^b_DENSITY.*", regex = TRUE) |>
  exp() |>
  ## equivalence_test(range = c(0.755, 1.32))
  rope(range = c(0.755, 1.32))
# Proportion of samples inside the ROPE [0.76, 1.32]:

Parameter                | inside ROPE
--------------------------------------
fSEASONSummer            |      0.00 %
fSEASONAutumn            |     12.36 %
fSEASONWinter            |      5.83 %
DENSITYLow               |     74.44 %
fSEASONSummer:DENSITYLow |     12.08 %
fSEASONAutumn:DENSITYLow |     57.58 %
fSEASONWinter:DENSITYLow |     20.58 %
quinn.mcmc <-
  quinn.brmsNB |>
  as_draws_df("^b_fSEASON.*|^b_DENSITY.*", regex = TRUE) |>
  exp()
quinn.mcmc |>
  rope(range = c(0.755, 1.32))
# Proportion of samples inside the ROPE [0.76, 1.32]:

Parameter                | inside ROPE
--------------------------------------
fSEASONSummer            |      0.00 %
fSEASONAutumn            |     12.36 %
fSEASONWinter            |      5.83 %
DENSITYLow               |     74.44 %
fSEASONSummer:DENSITYLow |     12.08 %
fSEASONAutumn:DENSITYLow |     57.58 %
fSEASONWinter:DENSITYLow |     20.58 %
## note, the following is not quit correct, it does not get the CI correct
quinn.mcmc |>
  rope(range = c(0.755, 1.32)) |>
  plot(data = quinn.brmsNB)

quinn.mcmc |>
  equivalence_test(range = c(0.755, 1.32))
# Test for Practical Equivalence

  ROPE: [0.76 1.32]

Parameter                |        H0 | inside ROPE |      95% HDI
-----------------------------------------------------------------
fSEASONSummer            |  Rejected |      0.00 % | [2.46, 6.50]
fSEASONAutumn            | Undecided |     12.36 % | [1.05, 2.86]
fSEASONWinter            | Undecided |      5.83 % | [0.29, 0.89]
DENSITYLow               | Undecided |     74.44 % | [0.56, 1.57]
fSEASONSummer:DENSITYLow | Undecided |     12.08 % | [0.28, 1.03]
fSEASONAutumn:DENSITYLow | Undecided |     57.58 % | [0.47, 2.18]
fSEASONWinter:DENSITYLow | Undecided |     20.58 % | [0.21, 1.40]
quinn.brmsNB |> modelsummary(
  statistic = c("conf.low", "conf.high"),
  shape = term ~ statistic,
  exponentiate = TRUE
)
Warning: 
`modelsummary` uses the `performance` package to extract goodness-of-fit
statistics from models of this class. You can specify the statistics you wish
to compute by supplying a `metrics` argument to `modelsummary`, which will then
push it forward to `performance`. Acceptable values are: "all", "common",
"none", or a character vector of metrics names. For example: `modelsummary(mod,
metrics = c("RMSE", "R2")` Note that some metrics are computationally
expensive. See `?performance::performance` for details.
 This warning appears once per session.
(1)
Est. 2.5 % 97.5 %
b_Intercept 11.135 7.656 16.948
b_fSEASONSummer 4.122 2.456 6.502
b_fSEASONAutumn 1.754 1.045 2.861
b_fSEASONWinter 0.504 0.286 0.892
b_DENSITYLow 0.953 0.561 1.573
b_fSEASONSummer × DENSITYLow 0.523 0.280 1.029
b_fSEASONAutumn × DENSITYLow 0.989 0.475 2.179
b_fSEASONWinter × DENSITYLow 0.534 0.208 1.403
Num.Obs. 42
R2 0.727
ELPD -148.4
ELPD s.e. 6.1
LOOIC 296.8
LOOIC s.e. 12.1
WAIC 295.9
RMSE 7.54
quinn.brmsNB |> modelplot(coef_omit = "shape", exponentiate = TRUE)

11 Further investigations

## fold scale
quinn.rstanarmNB |>
  emmeans(~ DENSITY | fSEASON, type = "response") |>
  pairs()
fSEASON = Spring:
 contrast   ratio lower.HPD upper.HPD
 High / Low 0.892     0.311      1.67

fSEASON = Summer:
 contrast   ratio lower.HPD upper.HPD
 High / Low 2.175     1.033      3.77

fSEASON = Autumn:
 contrast   ratio lower.HPD upper.HPD
 High / Low 1.051     0.469      2.03

fSEASON = Winter:
 contrast   ratio lower.HPD upper.HPD
 High / Low 2.106     0.457      5.64

Point estimate displayed: median 
Results are back-transformed from the log scale 
HPD interval probability: 0.95 
## absolute response scale
quinn.rstanarmNB |>
  emmeans(~ DENSITY | fSEASON, type = "link") |>
  regrid() |>
  pairs()
fSEASON = Spring:
 contrast   estimate lower.HPD upper.HPD
 High - Low   -1.233    -9.768      7.24

fSEASON = Summer:
 contrast   estimate lower.HPD upper.HPD
 High - Low   25.594     4.111     53.14

fSEASON = Autumn:
 contrast   estimate lower.HPD upper.HPD
 High - Low    0.924   -13.659     17.31

fSEASON = Winter:
 contrast   estimate lower.HPD upper.HPD
 High - Low    2.881    -0.856      8.00

Point estimate displayed: median 
HPD interval probability: 0.95 
quinn.em <- quinn.rstanarmNB |>
  emmeans(~ DENSITY | fSEASON, type = "link") |>
  pairs() |>
  gather_emmeans_draws() |>
  mutate(Fit = exp(.value))
head(quinn.em)
# A tibble: 6 × 7
# Groups:   contrast, fSEASON [1]
  contrast   fSEASON .chain .iteration .draw  .value   Fit
  <fct>      <fct>    <int>      <int> <int>   <dbl> <dbl>
1 High - Low Spring      NA         NA     1 -0.138  0.871
2 High - Low Spring      NA         NA     2 -0.0968 0.908
3 High - Low Spring      NA         NA     3  0.559  1.75 
4 High - Low Spring      NA         NA     4  0.113  1.12 
5 High - Low Spring      NA         NA     5  0.194  1.21 
6 High - Low Spring      NA         NA     6 -0.122  0.885
g2 <- quinn.em |>
  group_by(contrast, fSEASON) |>
  median_hdci() |>
  ggplot() +
  geom_vline(xintercept = 1, linetype = "dashed") +
  geom_pointrange(aes(x = Fit, y = fSEASON, xmin = Fit.lower, xmax = Fit.upper)) +
  scale_x_continuous("Effect size (High/Low)", trans = scales::log2_trans(), breaks = unique(as.vector(2^(0:4 %o% c(-1, 1))))) +
  theme_classic()
g2

ggplot(quinn.em, aes(x = Fit)) +
  geom_histogram() +
  geom_vline(xintercept = 1, linetype = "dashed") +
  scale_x_continuous("Effect size (High/Low)", trans = scales::log2_trans(), breaks = unique(as.vector(2^(0:4 %o% c(-1, 1))))) +
  facet_wrap(fSEASON ~ contrast, scales = "free")
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

quinn.em |>
  group_by(contrast, fSEASON) |>
  median_hdci(Fit)
# A tibble: 4 × 8
  contrast   fSEASON   Fit .lower .upper .width .point .interval
  <fct>      <fct>   <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
1 High - Low Spring  0.892  0.361   1.73   0.95 median hdci     
2 High - Low Summer  2.17   1.03    3.77   0.95 median hdci     
3 High - Low Autumn  1.05   0.460   2.02   0.95 median hdci     
4 High - Low Winter  2.11   0.457   5.64   0.95 median hdci     
# Probability of effect
quinn.em |>
  group_by(contrast, fSEASON) |>
  summarize(P = mean(Fit > 1))
`summarise()` has grouped output by 'contrast'. You can override using the
`.groups` argument.
# A tibble: 4 × 3
# Groups:   contrast [1]
  contrast   fSEASON     P
  <fct>      <fct>   <dbl>
1 High - Low Spring  0.364
2 High - Low Summer  0.988
3 High - Low Autumn  0.546
4 High - Low Winter  0.926
## Probability of effect greater than 10%
quinn.em |>
  group_by(contrast, fSEASON) |>
  summarize(P = mean(Fit > 1.1))
`summarise()` has grouped output by 'contrast'. You can override using the
`.groups` argument.
# A tibble: 4 × 3
# Groups:   contrast [1]
  contrast   fSEASON     P
  <fct>      <fct>   <dbl>
1 High - Low Spring  0.277
2 High - Low Summer  0.981
3 High - Low Autumn  0.451
4 High - Low Winter  0.9  
## Using summarise_draws
quinn.rstanarmNB |>
  emmeans(~ DENSITY | fSEASON, type = "link") |>
  pairs() |>
  tidy_draws() |>
  exp() |>
  summarise_draws(median,
    HDInterval::hdi,
    P = ~ mean(.x > 1)
  )
# A tibble: 4 × 5
  variable                           median lower upper     P
  <chr>                               <dbl> <dbl> <dbl> <dbl>
1 contrast High - Low fSEASON Spring  0.892 0.311  1.67 0.364
2 contrast High - Low fSEASON Summer  2.17  1.03   3.77 0.988
3 contrast High - Low fSEASON Autumn  1.05  0.469  2.03 0.546
4 contrast High - Low fSEASON Winter  2.11  0.457  5.64 0.926
newdata <- with(quinn, expand.grid(
  fSEASON = levels(fSEASON),
  DENSITY = levels(DENSITY)
))
Xmat <- model.matrix(~ fSEASON * DENSITY, data = newdata)
as.matrix(quinn.rstanarmNB) |> head()
          parameters
iterations (Intercept) fSEASONSummer fSEASONAutumn fSEASONWinter  DENSITYLow
      [1,]    2.381146      1.603036    0.22101808    -0.6949756  0.13761188
      [2,]    2.226240      1.278036    0.95572540    -0.5233653  0.09676273
      [3,]    2.553322      1.604023    0.51357944    -0.8898770 -0.55918503
      [4,]    2.462228      1.722616    0.38875020    -0.8272478 -0.11262107
      [5,]    2.387923      1.023239    0.04979926    -0.6542586 -0.19358063
      [6,]    2.279176      1.606230    0.64332711    -0.7916412  0.12224754
          parameters
iterations fSEASONSummer:DENSITYLow fSEASONAutumn:DENSITYLow
      [1,]               -1.1843259               0.40104431
      [2,]               -0.8951184              -0.50843028
      [3,]               -0.4686581               0.06325034
      [4,]               -1.1148374              -0.10442549
      [5,]               -0.2206245               0.84074413
      [6,]               -0.5920347              -0.42434611
          parameters
iterations fSEASONWinter:DENSITYLow reciprocal_dispersion
      [1,]              -0.21297875              3.692986
      [2,]              -1.03336169              4.445364
      [3,]              -0.10664788              5.414934
      [4,]              -0.61580878              5.067576
      [5,]              -0.76853431              2.328569
      [6,]              -0.04352769              3.060953
## coefs <- as.matrix(quinn.rstanarmNB)
coefs <- as.matrix(as.data.frame(quinn.rstanarmNB) |>
  dplyr:::select(-reciprocal_dispersion)) |>
  as.matrix()
fit <- exp(coefs %*% t(Xmat))
newdata <- newdata |>
  cbind(tidyMCMC(fit, conf.int = TRUE, conf.method = "HPDinterval"))
head(newdata)
  fSEASON DENSITY term  estimate std.error  conf.low conf.high
1  Spring    High    1 10.508575  2.723153  5.696427  15.90726
2  Summer    High    2 49.905997 11.822240 30.040811  72.23297
3  Autumn    High    3 20.430422  5.016132 11.575267  30.16747
4  Winter    High    4  5.915299  1.703658  3.054870   9.15440
5  Spring     Low    5 11.858948  3.351021  6.102056  18.10083
6  Summer     Low    6 23.176212  5.651388 14.416983  35.28266
ggplot(newdata, aes(y = estimate, x = fSEASON, fill = DENSITY)) +
  geom_blank() +
  geom_line(aes(x = as.numeric(fSEASON), ymin = conf.low, ymax = conf.high, linetype = DENSITY),
    position = position_dodge(0.2)
  ) +
  geom_pointrange(aes(ymin = conf.low, ymax = conf.high),
    shape = 21,
    position = position_dodge(0.2)
  )
Warning in geom_line(aes(x = as.numeric(fSEASON), ymin = conf.low, ymax =
conf.high, : Ignoring unknown aesthetics: ymin and ymax

# Compare high and low in each season
# via contrasts
newdata <- with(quinn, expand.grid(
  fSEASON = levels(fSEASON),
  DENSITY = levels(DENSITY)
))
## factor differences
Xmat <- model.matrix(~ fSEASON * DENSITY, data = newdata)
Xmat.high <- Xmat[newdata$DENSITY == "High", ]
Xmat.low <- Xmat[newdata$DENSITY == "Low", ]
Xmat.density <- Xmat.high - Xmat.low
rownames(Xmat.density) <- levels(quinn$fSEASON)
coefs <- as.matrix(as.data.frame(quinn.rstanarmNB) |> dplyr:::select(-reciprocal_dispersion))
fit <- exp(coefs %*% t(Xmat.density))
tidyMCMC(fit, conf.int = TRUE, conf.method = "HPDinterval")
# A tibble: 4 × 5
  term   estimate std.error conf.low conf.high
  <chr>     <dbl>     <dbl>    <dbl>     <dbl>
1 Spring    0.952     0.358    0.311      1.67
2 Summer    2.27      0.743    1.03       3.77
3 Autumn    1.13      0.433    0.469      2.03
4 Winter    2.53      1.55     0.457      5.64
## or absolute
fit.high <- coefs %*% t(Xmat.high)
fit.low <- coefs %*% t(Xmat.low)
fit <- exp(fit.high) - exp(fit.low)
# fit = exp(fit.high - fit.low)
tidyMCMC(fit, conf.int = TRUE, conf.method = "HPDinterval")
# A tibble: 4 × 5
  term  estimate std.error conf.low conf.high
  <chr>    <dbl>     <dbl>    <dbl>     <dbl>
1 1       -1.35       4.32   -9.77       7.24
2 2       26.7       13.1     4.11      53.1 
3 3        0.700      7.97  -13.7       17.3 
4 4        2.96       2.23   -0.856      8.00
quinn.brmsNB |>
  emmeans(~ DENSITY | fSEASON, type = "response") |>
  pairs()
Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.
fSEASON = Spring:
 contrast   ratio lower.HPD upper.HPD
 High / Low  1.05     0.599      1.66

fSEASON = Summer:
 contrast   ratio lower.HPD upper.HPD
 High / Low  2.00     1.043      3.10

fSEASON = Autumn:
 contrast   ratio lower.HPD upper.HPD
 High / Low  1.06     0.571      1.78

fSEASON = Winter:
 contrast   ratio lower.HPD upper.HPD
 High / Low  1.98     0.686      4.29

Point estimate displayed: median 
Results are back-transformed from the log scale 
HPD interval probability: 0.95 
## absolute response scale
quinn.brmsNB |>
  emmeans(~ DENSITY | fSEASON, type = "link") |>
  regrid() |>
  pairs()
Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.
fSEASON = Spring:
 contrast   estimate lower.HPD upper.HPD
 High - Low    0.539    -5.303      6.19

fSEASON = Summer:
 contrast   estimate lower.HPD upper.HPD
 High - Low   22.574     3.322     40.96

fSEASON = Autumn:
 contrast   estimate lower.HPD upper.HPD
 High - Low    1.152    -9.736     11.96

fSEASON = Winter:
 contrast   estimate lower.HPD upper.HPD
 High - Low    2.735    -0.765      6.31

Point estimate displayed: median 
HPD interval probability: 0.95 
quinn.em <- quinn.brmsNB |>
  emmeans(~ DENSITY | fSEASON, type = "link") |>
  pairs() |>
  gather_emmeans_draws() |>
  mutate(Fit = exp(.value))
Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.
head(quinn.em)
# A tibble: 6 × 7
# Groups:   contrast, fSEASON [1]
  contrast   fSEASON .chain .iteration .draw .value   Fit
  <fct>      <fct>    <int>      <int> <int>  <dbl> <dbl>
1 High - Low Spring      NA         NA     1  0.170  1.19
2 High - Low Spring      NA         NA     2  0.314  1.37
3 High - Low Spring      NA         NA     3  1.02   2.76
4 High - Low Spring      NA         NA     4  0.797  2.22
5 High - Low Spring      NA         NA     5  0.299  1.35
6 High - Low Spring      NA         NA     6  0.224  1.25
g2 <- quinn.em |>
  group_by(contrast, fSEASON) |>
  median_hdci() |>
  ggplot() +
  geom_vline(xintercept = 1, linetype = "dashed") +
  geom_pointrange(aes(x = Fit, y = fSEASON, xmin = Fit.lower, xmax = Fit.upper)) +
  scale_x_continuous("Effect size (High/Low)", trans = scales::log2_trans(), breaks = unique(as.vector(2^(0:4 %o% c(-1, 1))))) +
  theme_classic()
g2

ggplot(quinn.em, aes(x = Fit)) +
  geom_histogram() +
  geom_vline(xintercept = 1, linetype = "dashed") +
  scale_x_continuous("Effect size (High/Low)", trans = scales::log2_trans(), breaks = unique(as.vector(2^(0:4 %o% c(-1, 1))))) +
  facet_wrap(fSEASON ~ contrast, scales = "free")
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

quinn.em |>
  group_by(contrast, fSEASON) |>
  median_hdci(Fit)
# A tibble: 4 × 8
  contrast   fSEASON   Fit .lower .upper .width .point .interval
  <fct>      <fct>   <dbl>  <dbl>  <dbl>  <dbl> <chr>  <chr>    
1 High - Low Spring   1.05  0.599   1.66   0.95 median hdci     
2 High - Low Summer   2.00  1.08    3.15   0.95 median hdci     
3 High - Low Autumn   1.06  0.571   1.78   0.95 median hdci     
4 High - Low Winter   1.98  0.686   4.29   0.95 median hdci     
# Probability of effect
quinn.em |>
  group_by(contrast, fSEASON) |>
  summarize(P = mean(Fit > 1))
`summarise()` has grouped output by 'contrast'. You can override using the
`.groups` argument.
# A tibble: 4 × 3
# Groups:   contrast [1]
  contrast   fSEASON     P
  <fct>      <fct>   <dbl>
1 High - Low Spring  0.583
2 High - Low Summer  0.995
3 High - Low Autumn  0.585
4 High - Low Winter  0.933
## Probability of effect greater than 10%
quinn.em |>
  group_by(contrast, fSEASON) |>
  summarize(P = mean(Fit > 1.1))
`summarise()` has grouped output by 'contrast'. You can override using the
`.groups` argument.
# A tibble: 4 × 3
# Groups:   contrast [1]
  contrast   fSEASON     P
  <fct>      <fct>   <dbl>
1 High - Low Spring  0.435
2 High - Low Summer  0.986
3 High - Low Autumn  0.453
4 High - Low Winter  0.901

12 Summary figures

newdata <- quinn.rstanarmNB |>
  emmeans(~ fSEASON | DENSITY, type = "response") |>
  as.data.frame()
head(newdata)
 fSEASON DENSITY     prob lower.HPD upper.HPD
 Spring  High    10.11729  5.696427  15.90726
 Summer  High    48.05665 30.040811  72.23297
 Autumn  High    19.84166 11.575267  30.16747
 Winter  High     5.69321  3.054870   9.15440
 Spring  Low     11.38619  6.102056  18.10083
 Summer  Low     22.40605 14.416983  35.28266

Point estimate displayed: median 
Results are back-transformed from the log scale 
HPD interval probability: 0.95 
g1 <- ggplot(newdata, aes(y = prob, x = fSEASON, color = DENSITY)) +
  geom_pointrange(aes(ymin = lower.HPD, ymax = upper.HPD),
    position = position_dodge(width = 0.2)
  ) +
  theme_classic()
g1 + g2

newdata <- quinn.brmsNB %>%
  emmeans(~ fSEASON | DENSITY, type = "response") |>
  as.data.frame()
Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.
head(newdata)
 fSEASON DENSITY     prob lower.HPD upper.HPD
 Spring  High    11.13501  7.322182  16.08760
 Summer  High    45.59039 31.359375  64.08547
 Autumn  High    19.42992 13.214126  27.35445
 Winter  High     5.63679  3.465218   8.44709
 Spring  Low     10.66957  6.686198  15.37904
 Summer  Low     22.68431 15.127838  32.66172

Point estimate displayed: median 
Results are back-transformed from the log scale 
HPD interval probability: 0.95 
g1 <- ggplot(newdata, aes(y = prob, x = fSEASON, color = DENSITY)) +
  geom_pointrange(aes(ymin = lower.HPD, ymax = upper.HPD),
    position = position_dodge(width = 0.2)
  ) +
  theme_classic()
g1 + g2

13 Observation-level random effects

13.1 brms

quinn <- quinn |>
  group_by(fSEASON, DENSITY) |>
  mutate(Obs = factor(1:n()))

quinn.form <- bf(RECRUITS ~ fSEASON * DENSITY + (1 | Obs), family = poisson(link = "log"))
get_prior(quinn.form, data = quinn)
                  prior     class                     coef group resp dpar
                 (flat)         b                                         
                 (flat)         b               DENSITYLow                
                 (flat)         b            fSEASONAutumn                
                 (flat)         b fSEASONAutumn:DENSITYLow                
                 (flat)         b            fSEASONSummer                
                 (flat)         b fSEASONSummer:DENSITYLow                
                 (flat)         b            fSEASONWinter                
                 (flat)         b fSEASONWinter:DENSITYLow                
 student_t(3, 2.6, 2.5) Intercept                                         
   student_t(3, 0, 2.5)        sd                                         
   student_t(3, 0, 2.5)        sd                            Obs          
   student_t(3, 0, 2.5)        sd                Intercept   Obs          
 nlpar lb ub       source
                  default
             (vectorized)
             (vectorized)
             (vectorized)
             (vectorized)
             (vectorized)
             (vectorized)
             (vectorized)
                  default
        0         default
        0    (vectorized)
        0    (vectorized)
quinn.brmsU <- brm(quinn.form,
  data = quinn,
  refresh = 0,
  chains = 3,
  iter = 5000,
  thin = 5,
  warmup = 2000
)
Compiling Stan program...
Start sampling
preds <- posterior_predict(quinn.brmsU, nsamples = 250, summary = FALSE)
Warning: Argument 'nsamples' is deprecated. Please use argument 'ndraws'
instead.
quinn.resids <- createDHARMa(
  simulatedResponse = t(preds),
  observedResponse = quinn$RECRUITS,
  fittedPredictedResponse = apply(preds, 2, median),
  integerResponse = TRUE
)
plot(quinn.resids)

newdata <- emmeans(quinn.brmsU, ~ fSEASON | DENSITY, type = "response") |> as.data.frame()
Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.
newdata
DENSITY = High:
 fSEASON     rate lower.HPD upper.HPD
 Spring   9.76608   6.61843  13.61163
 Summer  47.15867  34.03303  62.28887
 Autumn  19.23758  12.81330  25.54698
 Winter   5.50793   3.22782   7.96708

DENSITY = Low:
 fSEASON     rate lower.HPD upper.HPD
 Spring  10.84652   7.23020  15.45789
 Summer  21.49778  14.60513  28.04147
 Autumn  16.29993  10.60382  22.41411
 Winter   2.39485   0.90701   4.50590

Point estimate displayed: median 
Results are back-transformed from the log scale 
HPD interval probability: 0.95 
ggplot(newdata, aes(y = rate, x = fSEASON, color = DENSITY)) +
  geom_pointrange(aes(ymin = lower.HPD, ymax = upper.HPD),
    position = position_dodge(width = 0.2)
  )

14 References

Quinn, G. P. 1988. “Ecology of the Intertidal Pulmonate Limpet Siphonaria Diemenensis Quoy Et Gaimard. II Reproductive Patterns and Energetics.” Journalofexperimentalmarinebiologyandecology 117: 137–56.